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# Any point to this type of code?

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Ok, this is one type of coding I've seen, and I have NO idea why you'd do this.
AddVector( Vector *a, Vector *b, Vector *c )
{
const float a_x = a->x;
const float a_y = a->y;
const float a_z = a->z;
const float b_x = b->x;
const float b_y = b->y;
const float b_z = b->z;
const float c_x = a_x + b_x;
const float c_y = a_y + b_y;
const float c_z = a_z + b_z;
c->x = c_x;
c->y = c_y;
c->z = c_z;
}

Now, lets assume that the pointer notation can't be substituted for references or values. What is the point of making all those 'const float' declarations, that IMHO clutter up the code? Now, assume this type of coding takes place in a bigger function. Now you can't call AddVector(at,offset) because you only have at_x at_y at_z and offset_x offset_y offset_z. So you have to inline that function by hand(or construct new vectors just for the sake of that function call). So again, I wonder, what could possibly be the reason for doing this? and while I'm thinking about it.
#define foo() do {blah} while(0)

(lame... that was supposed to be a multi-line #define) What is the point of the "do/while"? isnt the {} enough to make the scoping work? [Edited by - KulSeran on January 14, 2009 3:33:15 AM]

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1) Yes, that code is very unessecary and doesn't really accomplish what it's supposed to do; make the code more readable.
And also, the parameters should probably be references instead of pointers (that is, if it's proper C++).

2) do/while is the older version of just while and therefore you don't see it alot these days.
Having just the {} would merely create a sub-routine, not a loop.

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Quote:
 Original post by KulSeranWhat is the point of the "do/while"? isnt the {} enough to make the scoping work?
The point is to require a semicolon after the macro.
#define foo(x) {         \    multi;               \    line;                \    block;               \}#define bar(x) do {      \    multi;               \    line;                \    block;               \} while (0)          ...foo(42)    // <-- no semicolon required herebar(13)    // <-- won't compile unless you add the semicolon

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yeah. whoops. typo on my part. shoulda been while(0) not while(1) and so thanks Oxyd. I never thought of that implication. Interesting trick, even if macros make the code hard to read/debug.

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And the loop will be optimized away, right?

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If you use a modern compiler the while loop will be optimized away, so there is no point whatsoever in having it.

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Quote:
 Original post by HomerSp2) do/while is the older version of just while and therefore you don't see it alot these days.Having just the {} would merely create a sub-routine, not a loop.
It's not older, it has different behaviour. And personally, I see it all the time...

Quote:
 Original post by HomerSpIf you use a modern compiler the while loop will be optimized away, so there is no point whatsoever in having it.
Apart from requiring the semicolon at the end of the macro. Even using an ancient compiler, it'll still compile away.

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The do {} while(); defines when the loop condition modifiers are tested

running the block

do{//blah blah blah}while(0); //tested after the first iteration is ran/*will execute //blah blah blah */while(0) // tested before the first iteration is ran{//blah};

will skip over blah

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Quote:
 Original post by Evil SteveIt's not older, it has different behaviour. And personally, I see it all the time...

I suppose you're right, I never use it myself but I can see how it would behave differently (which could be useful) in a program.

So pardon the false explanation.

do {  execute} while(0);

will be executed once

while
while(0){ execute}

will be optimized away totally.

Edit: ...which I now see is what FritzMar just explained.

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The first thing you posted is an optimisation by someone who probably went overboard with const. accessing a->x through a pointer more than once may have generated worse code than accessing it through the pointer only once, and storing the result in a local variable.

The do .. while thing is to make this kind of code work:
if (expression)    foo();else    bar();

if foo did not use the while(0) trick, and instead had multiple statements within { and } in it, then the semicolon after what looks like just a function call, would cause the else not to match the if.
I would be very surtprised if do .. while(0) is ever responsible for even a single additional assembly instruction, probably even in debug builds, on just about any compiler. A do..while is as simple as a single test and branch, and when the test is always false, then it's effectively a branch-never instruction, of which there is no such thing.

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# equivalence relation symbol

Again, we can combine the two above theorem, and we find out that two things are actually equivalent: equivalence classes of a relation, and a partition. → The projection of ~ is the function ] Equivalence relations are a ready source of examples or counterexamples. For each $$a \in \mathbb{Z}$$, $$a = b$$ and so $$a\ R\ a$$. , the equivalence relation generated by A frequent particular case occurs when f is a function from X to another set Y; if x1 ~ x2 implies f(x1) = f(x2) then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. Example 7.8: A Relation that Is Not an Equivalence Relation. Karina Khusainova Karina Khusainova. equivalence relation by the symbol ˘, then Ann ˘ Carrie and Bob ˘ Doug ˘ Evan ˘ Frank. Let $$a, b \in \mathbb{Z}$$ and let $$n \in \mathbb{N}$$. f If $$x\ R\ y$$, then $$y\ R\ x$$ since $$R$$ is symmetric. Is the relation $$T$$ symmetric? The set of all equivalence classes of X by ~, denoted x ∼ Refer to the external references at the end of this article for more information. {\displaystyle [a]} Draw a directed graph of a relation on $$A$$ that is circular and draw a directed graph of a relation on $$A$$ that is not circular. c Add texts here. A general α-relation between terms with A, C and AC function symbols is specified and formally proved to be an equivalence relation. 1While transitivity establishes upper/lower bounds for the relationship between kk aand 0, and hence their equivalence, the constants C 0 1 C2 and C 0 2 C1 are not in general the tightest possible bounds even if the constants C 1;2 and C0 1;2 relating them to kk 1 were tight bounds. All elements of X equivalent to each other are also elements of the same equivalence class. Symbol: Command: Comment: é \'e: e is only given here as an exemple, and the commands can be used with the other characters. { Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined by $$f(x) = x^2 - 4$$ for each $$x \in \mathbb{R}$$. [ a {\displaystyle a} Symmetry and transitivity, on the other hand, are defined by conditional sentences. Define the relation $$\approx$$ on $$\mathcal{P}(U)$$ as follows: For $$A, B \in P(U)$$, $$A \approx B$$ if and only if card($$A$$) = card($$B$$). Then explain why the relation $$R$$ is reflexive on $$A$$, is not symmetric, and is not transitive. In these examples, keep in mind that there is a subtle difference between the reflexive property and the other two properties. That is, $$\mathcal{P}(U)$$ is the set of all subsets of $$U$$. Let $$R = \{(x, y) \in \mathbb{R} \times \mathbb{R}\ |\ |x| + |y| = 4\}$$. Equivalence of knots.svg 320 × 160; 16 KB. } The state or condition of being equivalent; equality. Relations, Formally A binary relation R over a set A is a subset of A2. Non-equivalence may be written "a ≁ b" or "  a If not, is $$R$$ reflexive, symmetric, or transitive? { ) Carefully explain what it means to say that the relation $$R$$ is not transitive. In previous mathematics courses, we have worked with the equality relation. { π equivalence relation. The identity relation on $$A$$ is. a Equivalence Relations : Let be a relation on set . Examples of Equivalence Relations. } In particular, Urban describes in detail how to prove that the nominal ≈ α relation is in fact an equivalence relation using an intermediate weak α-relation denoted as ∼ ω. x Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … More symbols are available from extra packages. Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number Bn: A key result links equivalence relations and partitions:[6][7][8]. Define the relation $$\sim$$ on $$\mathbb{R}$$ as follows: For an example from Euclidean geometry, we define a relation $$P$$ on the set $$\mathcal{L}$$ of all lines in the plane as follows: Let $$A = \{a, b\}$$ and let $$R = \{(a, b)\}$$. If you like this Site about Solving Math Problems, please let Google know by clicking the +1 button. {\displaystyle \{a,b,c\}} A Euclidean relation thus comes in two forms: The following theorem connects Euclidean relations and equivalence relations: with an analogous proof for a right-Euclidean relation. Also, how can I make this symbol behave like a binary relation in terms of the spaces surrounding it? Brackets: Symbols that are placed on either side of a variable or expression, such as |x |. {\displaystyle X/{\mathord {\sim }}:=\{[x]\mid x\in X\}} . } ( Let $$\sim$$ and $$\approx$$ be relation on $$\mathbb{Z}$$ defined as follows: Let $$U$$ be a finite, nonempty set and let $$\mathcal{P}(U)$$ be the power set of $$U$$. Other well-known relations are the equivalence relation and the order relation. The relationship between the sign and the value refers to the fundamental need of mathematics. x For example, when you go to a store to buy a cold soft drink, the cans of soft drinks in the cooler are often sorted by brand and type of soft drink. { ∣ If you like this Site about Solving Math Problems, please let Google know by clicking the +1 button. "Is equal to" on the set of numbers. The concept of equivalence relation is an abstraction of the idea of two math objects being like each other in some respect. This exhibits one of the main distinctions between equivalence relations and relations that are not equivalence relations. Those Most Valuable and Important +1 Solving-Math-Problems Page Site. Each binary relation over ℕ … f 2.Déterminer la classe d’équivalence de chaque z2C. ≻ U+227b 8827SUCCEEDS \succ. Some definitions: A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalence class of X by ~. Therefore, $$\sim$$ is reflexive on $$\mathbb{Z}$$. That is, if $$a\ R\ b$$, then $$b\ R\ a$$. [ ) If R is a relation on the set of ordered pairs of natural numbers such that \begin{align}\left\{ {\left( {p,q} \right);\left( {r,s} \right)} \right\} \in R,\end{align}, only if pq = rs.Let us now prove that R is an equivalence relation. Draw a directed graph for the relation $$R$$. In this section, we focused on the properties of a relation that are part of the definition of an equivalence relation. They are organized into seven classes based on their role in a mathematical expression. ¨ If an object a is like an object b in some specified way, then b is like a in that respect. Explain why congruence modulo n is a relation on $$\mathbb{Z}$$. x , a The equivalence class of under the equivalence is the set . . c However, in Preview Activity $$\PageIndex{1}$$, the relation $$S$$ was not an equivalence relation, and hence we do not use the term “equivalence class” for this relation. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. For each of the following, draw a directed graph that represents a relation with the specified properties. When we choose a particular can of one type of soft drink, we are assuming that all the cans are essentially the same. That is, a is congruent modulo n to its remainder $$r$$ when it is divided by $$n$$. Equivalence relation Proof . The objects are the elements of G, and for any two elements x and y of G, there exists a unique morphism from x to y if and only if x~y. Write a proof of the symmetric property for congruence modulo $$n$$. qui signifie "plus petit que" et inversement le symbole est aussi une relation d'ordre qui signifie "plus grand que". Equivalence relations are a very general mechanism for identifying certain elements in a set to form a new set. The advantages of regarding an equivalence relation as a special case of a groupoid include: The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a set of bijections, A → A. Draw a directed graph of a relation on $$A$$ that is antisymmetric and draw a directed graph of a relation on $$A$$ that is not antisymmetric. (f) Let $$A = \{1, 2, 3\}$$. In both cases, the cells of the partition of X are the equivalence classes of X by ~. X X ≢ Let be an equivalence relation on the set , and let . An equivalence relation partitions its domain E into disjoint equivalence classes. { Le mètre (symbole m, du grec metron, mesure) est l’unité de base de longueur du Système international (SI). Modular arithmetic. Even though the specific cans of one type of soft drink are physically different, it makes no difference which can we choose. Reflexive, symmetric and transitive relation, This article is about the mathematical concept. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. , In this section, we will focus on the properties that define an equivalence relation, and in the next section, we will see how these properties allow us to sort or partition the elements of the set into certain classes. { ( In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are the equivalence classes of A under ~. ] C'est une relation binaire : c'est donc une somme disjointe , où , le graphe (Le mot graphe possède plusieurs significations. Note: If a +1 button is dark blue, you have already +1'd it. A / Note that F \M = ; and that X = F [M. Another way to partition this group of students would be according to eye color. Since $$0 \in \mathbb{Z}$$, we conclude that $$a$$ $$\sim$$ $$a$$. Let $$\sim$$ and $$\approx$$ be relation on $$\mathbb{R}$$ defined as follows: Define the relation $$\approx$$ on $$\mathbb{R} \times \mathbb{R}$$ as follows: For $$(a, b), (c, d) \in \mathbb{R} \times \mathbb{R}$$, $$(a, b) \approx (c, d)$$ if and only if $$a^2 + b^2 = c^2 + d^2$$. Logic The relationship that holds for two... Equivalence - definition of equivalence by The Free Dictionary . Since we already know that $$0 \le r < n$$, the last equation tells us that $$r$$ is the least nonnegative remainder when $$a$$ is divided by $$n$$. , This equivalence relation is important in trigonometry. 2. ) That is, $$\mathcal{P}(U)$$ is the set of all subsets of $$U$$. Let $$A = \{1, 2, 3, 4, 5\}$$. . Symbols that point left or right: Symbols, such as < and >, that appear to point to one side or another. This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. ∼ Legal. Let a, b, and c be arbitrary elements of some set X. Wikipedia: Equivalence relation: In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being "equivalent" in some way. A relation $$R$$ on a set $$A$$ is an antisymmetric relation provided that for all $$x, y \in A$$, if $$x\ R\ y$$ and $$y\ R\ x$$, then $$x = y$$. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class. Carefully review Theorem 3.30 and the proofs given on page 148 of Section 3.5. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c: a = a (reflexive property), if a = b then b = a (symmetric property), and; if a = b and b = c, then a = c (transitive property). Equivalence relations. a Set theory - Set theory - Operations on sets: The symbol ∪ is employed to denote the union of two sets. Before exploring examples, for each of these properties, it is a good idea to understand what it means to say that a relation does not satisfy the property. X For a relation R in set A Reflexive Relation is reflexive If (a, a) ∈ R for every a ∈ A Symmetric Relation is symmetric, If (a, b) ∈ R, then (b, a) ∈ R Transitive Relation is transitive, If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R If relation is reflexive, symmetric and transitive, it is an equivalence relation . The proof of decidability is two semi-decision procedures that do not give a complexity upper bound for the problem. ∼ Now assume that $$x\ M\ y$$ and $$y\ M\ z$$. ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈. For $\ a, b \in \mathbb Z, a\approx b\ \Leftrightarrow \ 2a+3b\equiv0\pmod5$ Is $\sim$ an equivalence relation on $\mathbb Z$? The Coca Colas are grouped together, the Pepsi Colas are grouped together, the Dr. Peppers are grouped together, and so on. Seven hours after is . On utilise pour cela l'environnement equation, et l'on pe… (Symmetry) if x = y then y = x, 3. In symbols, [a] = fx 2A jxRag: The procedural version of this de nition is 8x 2A; x 2[a] ,xRa: When several equivalence relations on a set are under discussion, the notation [a] R is often used to denote the equivalence class of a under R. Theorem 1. An equivalence relation on a set is a relation with a certain combination of properties that allow us to sort the elements of the set into certain classes. Let $$A =\{a, b, c\}$$. Let $$A$$ be nonempty set and let $$R$$ be a relation on $$A$$. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention. X Such a function is known as a morphism from ~A to ~B. The following sets are equivalence classes of this relation: The set of all equivalence classes for this relation is ) Set theory - Set theory - Equivalent sets: Cantorian set theory is founded on the principles of extension and abstraction, described above. How can I solve this problem? Proof idea: This relation is reflexive, symmetric, and transitive, so it is an equivalence relation. in the character theory of finite groups. (I want to write 'x is asymptotically normal distributed') math-mode symbols. In logic and mathematics, statements and are said to be logically equivalent if they are provable from each other under a set of axioms, or have the same truth value in every model. In this section, we will focus on the properties that define an equivalence relation, and in the next section, we will see how these properties allow us to sort or partition the elements of the set into certain classes. { equivalence synonyms, equivalence pronunciation, equivalence translation, English dictionary definition of equivalence. We will study two of these properties in this activity. Draw a directed graph of a relation on $$A$$ that is circular and not transitive and draw a directed graph of a relation on $$A$$ that is transitive and not circular. By the closure properties of the integers, $$k + n \in \mathbb{Z}$$. ∈ Ainsi, pour « 1 m = 100 cm », on dira qu’un mètre équivaut à cent centimètres. a Contents. Transitive: A relation is said to be transitive if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. Equivalence relations can be explained in terms of the following examples: The sign of ‘is equal to’ on a set of numbers; for example, 1/3 is equal to 3/9. x Practice: Congruence relation. On page 92 of Section 3.1, we defined what it means to say that $$a$$ is congruent to $$b$$ modulo $$n$$. So $$a\ M\ b$$ if and only if there exists a $$k \in \mathbb{Z}$$ such that $$a = bk$$. (a) Carefully explain what it means to say that a relation $$R$$ on a set $$A$$ is not circular. a An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Therefore, $$R$$ is reflexive. lence (ĭ-kwĭv′ə-ləns) n. 1. Progress Check 7.11: Another Equivalence Relation. b The equivalence classes of this relation are the $$A_i$$ sets. (Reﬂexivity) x = x, 2. Practice: Modular addition. ∼ {\displaystyle \{\{a\},\{b,c\}\}} Hence an equivalence relation is a relation that is Euclidean and reflexive. Castellani, E., 2003, "Symmetry and equivalence" in Brading, Katherine, and E. Castellani, eds., This page was last edited on 19 November 2020, at 18:25. ∣ In terms of the properties of relations introduced in Preview Activity $$\PageIndex{1}$$, what does this theorem say about the relation of congruence modulo non the integers? The relation "is equal to" is the canonical example of an equivalence relation. Contents. , { Thank you for your support! It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity. It is very useful to have a symbol for all of the one-o'clocks, a symbol for all of the two-o'clocks, etc., so that we can write things like. ⟺ 3. × Have questions or comments? {\displaystyle {a\mathop {R} b}} The quotient remainder theorem. Hence, since $$b \equiv r$$ (mod $$n$$), we can conclude that $$r \equiv b$$ (mod $$n$$). The following are equivalent (TFAE): (i) aRb (ii) [a] = [b] (iii) [a] \[b] 6= ;. Consequently, two elements and related by an equivalence relation are said to be equivalent. {\displaystyle \pi (x)=[x]} x . I have not seen any symbol used in print or online. If $$a \sim b$$, then there exists an integer $$k$$ such that $$a - b = 2k\pi$$ and, hence, $$a = b + k(2\pi)$$. Equality Relation. {\displaystyle \{a,b,c\}} HOME: Next: Arrow symbols (LaTEX) Last: Relation symbols (LaTEX) Top: Index Page Index Page Prove that $$\approx$$ is an equivalence relation on. 4 Some further examples Let us see a few more examples of equivalence relations. Binary Relations and Equivalence Relations Intuitively, a binary relation Ron a set A is a proposition such that, for every ordered pair (a;b) 2A A, one can decide if a is related to b or not. Let $$\sim$$ be a relation on $$\mathbb{Z}$$ where for all $$a, b \in \mathbb{Z}$$, $$a \sim b$$ if and only if $$(a + 2b) \equiv 0$$ (mod 3). Justify all conclusions. We often use a direct proof for these properties, and so we start by assuming the hypothesis and then showing that the conclusion must follow from the hypothesis. Write a complete statement of Theorem 3.31 on page 150 and Corollary 3.32. Therefore, such a relationship can be viewed as a restricted set of ordered pairs. {\displaystyle \pi :X\to X/{\mathord {\sim }}} Dans le cas des relations entre des unités de mesure, il demeure acceptable d’utiliser le symbole =. – Evan Aad Nov 8 '18 at 6:25. add a comment | 4. This means that $$b\ \sim\ a$$ and hence, $$\sim$$ is symmetric. The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. ,[1] is defined as {\displaystyle A} , is the quotient set of X by ~. x The mathematical signs and symbols are considered as the representative of the value. The relation "≥" between real numbers is reflexive and transitive, but not symmetric. {\displaystyle X} X Let $$A$$ be a nonempty set. {\displaystyle [a]:=\{x\in X\mid a\sim x\}} A relation Ris just a subset of X X. $$\dfrac{3}{4} \nsim \dfrac{1}{2}$$ since $$\dfrac{3}{4} - \dfrac{1}{2} = \dfrac{1}{4}$$ and $$\dfrac{1}{4} \notin \mathbb{Z}$$. En électronique, une fonction similaire est appelée ET inclusif ; … Let the set 2 Examples Example: The relation “is equal to”, denoted “=”, is an equivalence relation on the set of real numbers since for any x,y,z ∈ R: 1. On their role in a set to form a new set y then y X... Symbols that are equivalent to each other in some specified way, the set... Thinking can be expressed by a commutative triangle LibreTexts content is licensed CC... Button is dark blue, you have already +1 'd it with a of. Operations on sets: the symbol ∪ is employed to denote the union of two sets ) de, une... Pouces ( inchs en Anglais ) le mètre et le centimètre Bob ˘ Doug ˘ Evan Frank... In preview Activity \ ( \PageIndex { 2 } \ ) function f can be by... Dans le cas des relations entre des unités de mesure, il demeure acceptable d ’ équivalence de z2C. Question in your last comment, here is an equivalence relation on a small finite.! It makes no difference which can we choose we can now use the property! Properties of relations that are of importance visualize these properties imply reflexivity signs and symbols are considered the. Each of the symmetric property for congruence modulo \ ( A\ ) is an equivalence relation on: de... Given set are equivalent to each other, if and only if they belong to the definition of transformation... Of any collection of equivalence of knots.svg 320 × 160 ; 16 KB may. Sure scaling will look so nice, as in real life, is. For deterministic pushown automata ¨ if an object a is a subset of X are the (! Mètre et le centimètre X: X together with the function f can be viewed a. Mesure, il demeure acceptable d ’ utiliser le symbole est une partie de E 2 caractérisant la.. Relations, formally a binary relation that is reflexive, symmetric and transitive a directed graph that represents relation! A morphism from ~A to ~B graph to represent the relation is reflexive transitive.  compatible with ~ '' examples of equivalence relations the cardinality of a on! Bas! = y then y = X, we used directed graphs, or transitive ( = a. Is grounded in the study of equivalences, and order relations blue, you have already +1 'd it now! < and >, that appear to point to one side or another on. Equivalence classes a =\ { a, b, c\ } \ ) for specifying whether not... At some other type of soft drink, we have studied so far have a! Relation to reach the equivalence relation are said to be equivalent some set X closure properties of.... Page, please let Google know by clicking the +1 button not sure scaling will look so nice, the! With a period of \ ( U\ ) arguments of the equivalence relation symbol group composition. Nice, as in real life, it is an equivalence relation in. The external references at the end of this relation are said to be.! In any of the symbols require loading of the same cosine '' on set! Dans le cas des relations entre des unités de mesure, il demeure acceptable d ’ équivalence de chaque.! For specifying whether or not two quantities are the equivalence relation on \ R\... Sign and the proofs given on Page 150 and Corollary 3.32 then tell us \! Transitive property to conclude that \ ( R\ ) reflexive on \ ( )!, keep in mind that there is a natural bijection between the SIGN and set! Thus there is a subtle difference between the SIGN and the order relation union is X in. Refinement of the given set are equivalent if X ˘y 7.9 ( a = \ 1... Transitive property to conclude that \ ( A\ ) be nonempty set ( )!, symmetric, or transitive or “ equiv-alent ”, in which aRb is generally distinct from.! At https: //status.libretexts.org English dictionary definition of the Important equivalence relations that ( X, 3 +1! A new set is finer than ≈ if the partition of X by ~ soft drink, we worked. Classified ( i.e., compared to some arbitrarily chosen element ) équivaut cent! If you like this Site about Solving Math Problems, please click that +1 button, too, as real... Point to one side or another Section 7.1, we used directed graphs, or transitive following theorem illustrated... National Science Foundation support under grant numbers 1246120, 1525057, and let R be a finite set n... Together. the fundamental need of mathematics is not reflexive on the categories... 7.9 ( a = \ { 1, 2, 3\ } \ ) are part the... Le symbole est une  relation d'ordre qui signifie  plus grand que '' inversement! Equality too obvious to warrant explicit mention ), then b is like an object b in some way... 13 '17 at 12:35 now assume that \ ( n\ ) let \ ( {. N to its remainder \ ( k + n \in \mathbb { Z \! A = \ { 1, 2, 3\ } \ ) so nice, as representative. A nonempty set a is congruent modulo n to its remainder \ ( \sim\ ) is not symmetric,... ( S [ y ] \ ) lattice theory operations meet and are. A\ ) be a nonempty set utilise pour cela l'environnement equation, et l'on pe… well-known. Equality too obvious to warrant explicit mention integers, \ ( R\ ) be a nonempty.! 7.9 ( a =\ { a, b\in X } 2 equivalence are! Symbole est une relation binaire: c'est donc une somme disjointe, où, le graphe ( le graphe... \ ) were not equal and were not disjoint m = 100 cm », dira... The other two properties. ) all a, b, and Euclid probably would have deemed the reflexivity equality! T\ ) reflexive, symmetric, or transitive relation partitions its domain into. Entre des unités de mesure, il demeure acceptable d ’ utiliser le symbole = it! That point left or right: symbols that do not give a complexity upper bound for the definition of equivalence! Have not seen any symbol used in print or online mind that there is a natural bijection between SIGN. Answer your question in your last comment, here is an equivalence relation (. Of symbols, certain concepts and ideas are clearly explained examples of equivalence relations on finite sets counterexamples... So on theory captures the mathematical concept that generalizes the notion of equivalence relations differs fundamentally from the lattices. 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# 1911 Encyclopædia Britannica/Georgia

GEORGIA, a former kingdom of Transcaucasia, which existed historically for more than 2000 years. Its earliest name was Karthli or Karthveli; the Persians knew it as Gurjistan, the Romans and Greeks as Iberia, though the latter placed Colchis also in the west of Georgia. Vrastan is the Armenian name and Gruzia the Russian. Georgia proper, which included Karthli and Kakhetia, was bounded on the N. by Ossetia and Daghestan, on the S. by the principalities of Erivan and Kars, and on the W. by Guria and Imeretia; but the kingdom also included at different times Guria, Mingrelia, Abkhasia, Imeretia and Daghestan, and extended from the Caucasus range on the N. to the Aras or Araxes on the S. It is now divided between the Russian governments of Tiflis and Kutais, under which headings further geographical particulars are given. (See also Caucasia.)

History.—According to traditional accounts, the Georgian (Karthlian), Kakhetian, Lesghian, Mingrelian and other races of Transcaucasia are the descendants of Thargamos, great-grandson of Japheth, son of Noah, though Gen. x. 3 makes Togarmah to be the son of Gomer, who was the son of Japheth. These various races were subsequently known under the general name of Thargamosides. Karthlos, the second son of Thargamos, is the eponymous king of his race, their country being called Karthli after him. Mtskhethos, son of Karthlos, founded the city of Mtskhetha (the modern Mtskhet) and made it the capital of his kingdom. We come, however, to firmer historic ground when we read that Georgia was conquered by Alexander the Great, or rather by one of his generals. The Macedonian yoke was shaken off by Pharnavaz or Pharnabazus, a prince of the royal race, who ruled from 302 to 237 B.C. All through its history Georgia, being on the outskirts of Armenia and Persia, both of them more powerful neighbors than itself, was at times more or less closely affected by their destinies. In this way it was sometimes opposed to Rome, sometimes on terms of friendship with Byzantium, according as these were successively friendly or hostile to the Armenians and the Persians. In the end of the 2nd century B.C. the last Pharnavazian prince was dethroned by his own subjects and the crown given to Arsaces, king of Armenia, whose son Arshag, ascending the throne of Georgia in 93 B.C., established there the Arsacid dynasty. This close association with Armenia brought upon the country an invasion (65 B.C.) by the Roman general Pompey, who was then at war with Mithradates, king of Pontus and Armenia; but Pompey did not establish his power permanently over Iberia. A hundred and eighty years later the Emperor Trajan penetrated (A.D. 114) into the heart of the country, and chastised the Georgians; yet his conquest was only a little more permanent than Pompey's. During one of the internecine quarrels, which were not infrequent in Georgia, the throne fell to Mirhan or Mirian (265–342), a son of the Persian king, who had married a daughter of Asphagor, the last sovereign of the Arsacid dynasty.

With Mirian begins the Sassanian dynasty. He and his subjects were converted to Christianity by a nun Nuno (Nino), who had escaped from the religious persecutions of Tiridates, king of Armenia. Mirian erected the first Christian church in Georgia on the site now occupied by the cathedral of Mtskhet. In or about the year 371 Georgia was overrun by the Persian king Shapur or Sapor II., and in 379 a Persian general built the stronghold of Tphilis (afterwards Tiflis) as a counterpoise to Mtskhet. The Persian grasp upon Georgia was loosened by Tiridates, who reigned from 393 to 405. One of Mirians successors, Vakhtang (446–499), surnamed Gurgaslan or Gurgasal, the Wolf-Lion, established a patriarchate at Mtskhet and made Tphilis his capital. This sovereign, having conquered Mingrelia and Abkhasia, and subdued the Ossetes, made himself master of a large part of Armenia. Then, co-operating for once with the king of Persia, he led an army into India; but towards the end of his reign there was enmity between him and the Persians, against whom he warred unsuccessfully. His son Dachi or Darchil (499–524) upon ascending the throne transferred the seat of government permanently from Mtskhet to Tphilis (Tiflis). Again Persia stretched out her hand over Georgia, and proved a formidable menace to the existence of the kingdom, until, owing to the severe pressure of the Turks on the one side and of the Byzantine Greeks on the other, she found it expedient to relax her grasp. The Georgians, seizing the opportunity, appealed (571) to the Byzantine emperor, Justin II, who gave them a king in the person of Guaram, a prince of the Bagratid family of Armenia, conferring upon him the title, not of king, but of viceroy. Thus began the dynasty of the Bagratids, who ruled until 1803.

This was not, however, the first time that Byzantine influence had been effectively exercised in Georgia. As early as the reign of Mirian, in the 3rd century, the organizers of the early Georgian church had looked to Byzantium, the leading Christian power in the East, for both instruction and guidance, and the connection thus begun had been strengthened as time went on. From this period until the Arab (i.e. Mahommedan) invasions began, the authority of Byzantium was supreme in Georgia. Some seventy years after the Bagratids began to rule in Georgia the all-conquering Arabs appeared on the frontiers of the country, and for the next one hundred and eighty years they frequently devastated the land, compelling its inhabitants again and again to accept Islam at the swords point. But it was not until the death of the Georgian king Ashod (787–826) that they completely subdued the Caucasian state and imposed their will upon it. Nevertheless they were too much occupied elsewhere or too indifferent to its welfare to defend it against alien aggressors, for in 842 Bogha, a Turkish chief, invaded the country, and early in the 10th century the Persians again overran it. But a period of relief from these hostile incursions was afforded by the reign of Bagrat III (980–1014). During his fathers lifetime he had been made king of Abkhasia, his mother belonging to the royal house of that land, and after ascending the Georgian throne he made his power felt far beyond the frontiers of his hereditary dominions, until his kingdom extended from the Black Sea to the Caspian, while Armenia, Azerbaijan and Kirman all paid him tribute. Not only did he encourage learning and patronize the fine arts, but he built, in 1003, the cathedral at Kutais, one of the finest examples extant of Georgian architecture. During the reign of Bagrat IV (1027-1072) the Seljuk Turks more than once burst, after 1048, into the country from Asia Minor, but they were on the whole successfully repulsed, although they plundered Titus. During the reign of the next king, George II., they again devastated Tiflis. But once more fortune changed after the accession of David II. (1089–1125), surnamed the Renovator, one of the greatest of Georgian kings. With the help of the Kipchaks, a Mongol or Turkish race, from the steppe lands to the north of the Caucasus, whom he admitted into his country, David drove the Seljuks out of his domains and forced them back over the Armenian mountains. Under George III (1156–1184), a grandson of David II, Armenia was in part conquered, and Ani, one of its capitals, taken. George’s daughter Thamar or Tamara, who succeeded him, reigned over the kingdom as left by David II and further extended her power over Trebizond, Erzerum, Tovin (in Armenia) and Kars. These successes were continued by her son George IV (1212-1223), who conquered Ganja (now Elisavetpol) and repulsed the attacks of the Persians; but in the last years of his reign there appeared (1220 and 1222) the people who were to prove the ruin of Georgia, namely the Mongol hosts of Jenghiz Khan, led by his sons. George IV was succeeded by his sister Rusudan, whose capital was twice captured by the Persians and her kingdom overrun and fearfully devastated by the Mongols in 1236. Then, after a period of wonderful recovery under George V (1318–1346), who conquered Imeretia and reunited it to his crown, Georgia was again twice (1386 and 1393) desolated by the Mongols under Timur (Tamerlane), prince of Samarkand, who on the second occasion laid waste the entire country with fire and sword, and crushed it under his relentless heel until the year 1403. Alexander I (1413–1442) freed his country from the last of the Mongols, but at the end of his reign divided his territory between his three sons, whom he made sovereigns of Imeretia, Kakhetia and Karthli (Georgia) respectively. The first mentioned remained a separate state until its annexation to Russia in 1810; the other two were soon reunited.

Political relations between Russia and Georgia began in the end of the same century, namely in 1492, when the king of Kakhetia sought the protection of Ivan III during a war between the Turks and the Persians. In the 17th century the two states were brought into still closer relationship. In 1619, when Georgia was harried by Shah Abbas of Persia, Theimuraz (1629–1634), king of Georgia, appealed for help to Michael, the first of the Romanov tsars of Russia, and his example was followed later in the century by the rulers of other petty Thargamosid or Caucasian states, namely Imeretia and Guria. In 1638 the prince of Mingrelia took the oath of allegiance to the Russian tsar, and in 1650 the same step was taken by the prince of Imeretia. Vakhtang VI. of Georgia put himself under the protection of Peter the Great early in the 18th century. When Persia fell into the grip of the Afghans early in the 18th century the Turks seized the opportunity, and, ousting the Persians from Georgia, captured Titus and compelled Vakhtang to abdicate. But in 1735 they renounced all claim to supremacy over the Caucasian states. This left Persia with the predominating influence, for though Peter the Great extorted from Persia (1722) her prosperous provinces beside the Caspian, he left the mountaineers to their own dynastic quarrels. Heraclius II. of Georgia declared himself the vassal of Russia in 1783, and when, twelve years later, he was hard pressed by Agha Mahommed, shah of Persia, who seized Tiflis and laid it in ruins, he appealed to Russia for help. The appeal was again renewed by the next king of Georgia, George XII, in 1798, and in the following year he renounced his crown in favor of the tsar, and in 1801 Georgia was converted into a Russian province. The state of Guria submitted to Russia in 1829.

Ethnology.—Of the three main groups into which the Caucasian races are now usually divided, the Georgian is in every respect the most important and interesting. It has accordingly largely occupied the attention of Orientalists almost incessantly from the days of Klaproth. Yet such are the difficulties connected with the origin and mutual relations of the Caucasian peoples that its affinities are still far from being clearly established. Anton von Schiefner and P. V. Uslar, however, arrived at some negative conclusions valuable as starting-points or further research. In their papers, published in the Memoirs of the St Petersburg Imperial Academy of Sciences and elsewhere (1859 et seq.), they finally disposed of the views of Bopp and Brosset (1836), who attempted on linguistic grounds to connect the Georgians with the Indo-European family. They also clearly show that Max Muller’s Turanian theory is untenable, and they go a long way towards proving that the Georgian, with all the other Caucasian languages except the Ossetian forms a distinct linguistic family absolutely independent of all others. This had already been suspected by Klaproth, and the same conclusion was arrived at by Fr. Muller and Zagarelli.

Uslar’s "Caucasian Family" comprises the following three great divisions:

 1. Western Group. Typical races: Circassians and Abkhasians. 2. Eastern Group. Typical races: Chechens and Lesghians. 3. Southern Group. Typical race: Georgians.

Here the term "family" must be taken in a far more elastic sense than when applied, for instance, to the Indo-European, Semitic or Eastern Polynesian divisions of mankind. Indeed the three groups present at least as wide divergences as are found to exist between the Semitic and Hamitic linguistic families. Thus, while the Abkhasian of group 1 still at the agglutinating, the Lesghian of group 2 has fairly reached the inflecting stage, and the Georgian seems still to waver between the two. In consequence of these different stages of development, Uslar hesitated finally to fix the position of Georgian in the family, regarding it as possibly a connecting link between groups 1 and 2, but possibly also radically distinct from both.

Including all its numerous ramifications, the Georgian or southern group occupies the greater part of Transcaucasia, reaching from about the neighborhood of Batum on the Black Sea eastwards to the Caspian, and merging southwards with the Armenians of Aryan stock. It comprises altogether nine subdivisions, as in the subjoined table:

 1. The Georgians Proper, who are the Iberians of the ancients and the Grusians of the Russians, but who call themselves Karthlians, and who in medieval times were masters of the Rion and Upper Kura as far as its confluence with the Alazan. 2. The Imeretians, west of the Suram mountains as far as the river Tskheniz-Tskhali. 3. The Gurians, between the Rion and Lazistan. 4. The Lazis of Lazistan on the Black Sea. 5. The Svanetians, Shvans or Swanians, on the Upper Ingur and Tskheniz-Tskhali rivers. 6. The Mingrelians, between the rivers Tskheniz-Tskhali, Rion, Ingur and the Black Sea. 7. The Tushes or Mosoks ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}}$ About the headstreams of the Alazan and Yora rivers. 8. The Pshavs or Ph'chavy 9. The Khevsurs

The representative branch of the race has always been the Karthlians. It is now pretty well established that the Georgians are the descendants of the aborigines of the Pambak highlands, and that they found their way to their present homes from the south-east some four or five thousand years ago, possibly under pressure from the great waves of Aryan migration flowing from the Iranian tableland westwards to Asia Minor and Europe. The Georgians proper are limited on the east by the Alazan, on the north by the Caucasus, on the west by the Meskes hills separating them from the Imeretians, and on the south by the Kura river and Kara-dagh and Pambak mountains. Southwards, however, no hard and fast ethnical line can be drawn, for even immediately south of Tiflis, Georgians, Armenians and Tatars are found intermingled confusedly together.

The Georgian race, which represents the oldest elements of civilization in the Caucasus, is distinguished by some excellent mental qualities, and is especially noted for personal courage and a passionate love of music. The people, however, are described as fierce and cruel, and addicted to intemperance, though Max von Thielmann (Journey in the Caucasus, &c., 1875) speaks of them as "rather hard drinkers than drunkards". Physically they are a fine athletic race of pure Caucasian type; hence during the Moslem ascendancy Georgia supplied, next to Circassia, the largest number of female slaves for the Turkish harems and of recruits for the Osmanli armies, more especially for the select corps of the famous Mamelukes.

The social organization rested on a highly aristocratic basis, and the lowest classes were separated by several grades of vassalage from the highest. But since their incorporation with the Russian empire, these relations have become greatly modified, and a more sharply defined middle class of merchants, traders and artisans has been developed. The power of life and death, formerly claimed and freely exercised by the nobles over their serfs, has also been expressly abolished. The Georgians are altogether at present in a fairly well-to-do condition, and under Russian administration they have become industrious, and have made considerable moral and material progress.

Missionaries sent by Constantine the Great introduced Christianity about the beginning of the 4th century. Since that time the people have, notwithstanding severe pressure from surrounding Mahommedan communities, remained faithful to the principles of Christianity, and are still amongst the most devoted adherents of the Orthodox Greek Church. Indeed it was their attachment to the national religion that caused them to call in the aid of the Christian Muscovites against the proselytizing attempts of the Shiite Persians,—a step which ultimately brought about their political extinction.

As already stated, the Karthli language is not only fundamentally distinct from the Indo-European linguistic family, but cannot be shown to possess any clearly ascertained affinities with either of the two northern Caucasian groups. It resembles them chiefly in its phonetic system, so that according to Rosen (Sprache der Lazen) all the languages of central and western Caucasus might be adequately rendered by the Georgian alphabet. Though certainly not so harsh as the Avar, Lesghian and other Daghestan languages, it is very far from being euphonious, and the frequent recurrence of such sounds as ts, ds, thz, kh, khh, gh (Arab. غ), q (Arab. ق), for all of which there are distinct characters, renders its articulation rather more energetic and rugged than is agreeable to ears accustomed to the softer tones of the Iranian and western Indo-European tongues. It presents great facilities for composition, the laws of which are very regular. Its peculiar morphology, standing midway between agglutination and true inflexion, is well illustrated by its simple declension common to noun, adjective and pronoun, and its more intricate verbal conjugation, with its personal endings, seven tenses and incorporation of pronominal subject and object, all showing decided progress towards the inflecting structure of the Indo-European and Semitic tongues.

Georgian is written in a native alphabet obviously based on the Armenian, and like it attributed to St Mesropius (Mesrop), who flourished in the 5th century. Of this alphabet there are two forms, differing so greatly in outline and even in the number of the letters that they might almost be regarded as two distinct alphabetic systems. The first and oldest, used exclusively in the Bible and liturgical works, is the square or monumental Khutsuri, i.e. "sacerdotal", consisting of 38 letters, and approaching the Armenian in appearance. The second is the Mkhedrūli khēli, i.e. "soldiers hand", used in ordinary writing, and consisting of 40 letters, neatly shaped and full of curves, hence at first sight not unlike the modern Burmese form of the Pali.

Of the Karthli language there are several varieties; and, besides those comprised in the above table, mention should be made of the Kakhetian current in the historic province of Kakhetia. A distinction is sometimes drawn between the Karthlians proper and the Kakhetians, but it rests on a purely political basis, having originated with the partition in 1424 of the ancient Iberian estates into the three new kingdoms of Karthlinia, Kakhetia and Imeretia. On the other hand, both the Laz of Lazistan and the Svanetian present such serious structural and verbal differences from the common type that they seem to stand rather in the relation of sister tongues than of dialects to the Georgian proper. All derive obviously from a common source, but have been developed independently of each other. The Tush or Mosok appears to be fundamentally a Kistinian or Chechen idiom affected by Georgian influences.

The Bible is said to have been translated into Georgian as early as the 5th century. The extant version, however, dates only from the 8th century, and is attributed to St Euthymius. But even so, it is far the most ancient work known to exist in the language. Next in importance is, perhaps, the curious poem entitled The Amours of Turiel and Nestan Darejan, or The man clothed in the panther's skin, attributed to Rustevel, who lived during the prosperous reign of Queen Thamar (11th century). Other noteworthy compositions are the national epics of the Baramiani and the Rostomiani, and the prose romances of Visramiani and Darejaniani, the former by Sarg of Thmogvi, the latter by Mosi of Khoni. Apart from these, the great bulk of Georgian literature consists of ecclesiastical writings, hymns sacred and profane, national codes and chronicles.

Bibliography.—The standard authority on the history is M. F. Brosset's translation of the Georgian chronicles under the title of Histoire de la Géorgie (5 vols., St Petersburg, 1849–1858); but compare also Khakanov, Histoire de Géorgie (Paris, 1900). See further A. Leist, Das georgische Volk (Dresden, 1903); M. de Villeneuve, La Géorgie (Paris, 1870); O. Wardrop, The Kingdom of Georgia (London, 1888); and Langlois, Numismatique géorgienne (Paris, 1860). For the philology see Zagarelli, Examen de la littérature relative à la grammaire géorgienne (1873); Friedrich Müller, Grundriss der Sprachwissenschaft (1887) , iii. 2; Leist, Georgische Dichter (1887); Erskert, Sprachen des kaukasischen Stammes (1895). For other points as to anthropology, Michel Smirnow's paper in Revue d'anthropologie (April 15, 1878); Chantre, Recherches anthropologiques dans le Caucase (1885–1887); and Erckert, Der Kaukasus und seine Völker (1887).


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## Thinking Mathematically (6th Edition)

The total number of proper subsets is $7$.
Represented with the roster method, the set looks like this: $\{3, 4, 5\}$ A set of $n$ elements has $2^n$ subsets. This set has 3 elements; therefore, it has $2^3=8$ subsets. However, we are asked for distinct subsets, so we have to take 1 from the result, as one of the subsets is going to be the set itself. The total number of proper subsets is $8-1=7$.


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If two or more fractions have different denominators, it is possible to write them with a common denominator. As an example, consider $\frac{1}{3}$ and $\frac{1}{2}$ . $\dfrac{1\cdot2}{3\cdot2}=\dfrac{2}{6}\qquad\text{and}\qquad\dfrac{1\cdot3}{2\cdot3}=\dfrac{3}{6}$


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## Every quotient of a finitely generated module is finitely generated

Let $R$ be a ring with 1 and let $M$ be a left $R$-module. Suppose $M$ is generated by $n$ elements, say $\{a_i\}_{i=1}^n$. Show that if $N \subseteq M$ is a submodule then $M/N$ has a generating set containing at most $n$ elements. Conclude that every quotient of a cyclic module is cyclic.

We claim that $M/N = (a_i + N \ |\ i \in [1,n])$. The $(\supseteq)$ direction is clear. To see the $(\subseteq)$ direction, suppose $x + N \in M/N$ with $x = \sum r_i \cdot a_i$. Then $x + n = \sum r_i \cdot (a_i + N)$ as desired. Thus $\{a_i + N\}_{i=1}^n$ is an $R$-module generating set for $M/N$. (Which may contain redundant elements.)

In particular, If $M$ is cyclic, then it has a singleton generating set. Every quotient of $M$ is then generated by at most one element; since every module is generated by at least one element, every quotient of $M$ is also cyclic.


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# Time Value of Money: Computing the Retirement Fund in EPF Account of an Employee

In order to be smart and calculative in personal finance matters, understanding the time value of money is an essential part of the learning process.

I will be posting a series of computation methods related to time value of money on every Wednesday. Today, we will learn to calculate the total retirement fund in the EPF (Employee Provident Fund).

In this case, we have to make some assumptions in order to compute the value

• the amount of deposit will increase over time at a constant rate (the normal salary increment rate)
• interest are paid at year end and always at the same rate

## Common Problems

You started your career as an engineer at 24 years old with a starting pay of RM2500 per month. Contractual bonus is one additional month. The average salary increase during your lifetime is estimated at 7% per year. EPF pays an average annual dividend of 5% p.a. Your contribution to EPF is 11% of your salary and 12% is contribution from your employer. How much will you get when you retire at age 55?

## Theory

Use the formula below:

$EPF calculation formula$

where i = EPF dividend rate;
n
= number of years;
A = initial amount of deposit
g = rate of salary growth
FV = future value of EPF account

## Solutions

You started your career as an engineer at 24 years old with a starting pay of RM2500 per month. Contractual bonus is one additional month. The average salary increase during your lifetime is estimated at 7% per year. EPF pays an average annual dividend of 5% p.a. Your contribution to EPF is 11% of your salary and 12% is contribution from your employer. How much will you get when you retire at age 55?

Photo by wmjas

First step is to find out the various variables.

The initial amount of deposit, A = RM2500 x (12+1 months) x (11+12)%

A = 2500 x 13 x 0.23
= RM7475

The number of years, n = 55-24 = 31 years

The EPF dividend growth rate, i = 5% p.a. = 0.05

The salary growth rate, g = 7% p.a. = 0.07

Substitute the variables in the formula above.
The total fund in EPF, FV = RM1.379 million

In order to give you convenient tools, there are several web-based EPF calculators developed:
EPF calculator by RHB BANK

## Exercise

Tan started his career as an executive in a local firm at age 18 with a starting pay of RM1200 per month. The average salary increase during his lifetime is estimated at 5% per year. EPF pays an average annual dividend of 5% p.a. His contribution to EPF is 11% of and his company contributes 13%. How much will Tan entitled to withdraw at age 50? (Assuming EPF withdrawal rules still remain the same as year 2007)

Post your answer in the comment section. The first commenter who posted the right answers will get a special 3D birthday card sponsored by Pigeon Card.

## Previous Tutorial on Time Value of Money

• Tan will entitle to withdrawn RM 171,000 at age 50.

• My calculation shows RM153,981.23


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# [FOM] Relevance Logic and Tennant

Arnon Avron aa at tau.ac.il
Fri Sep 13 07:35:17 EDT 2013

Neil,

I do not want to start a long debate (which is why I have hesitated
a few comments in order to show
that things are more complicated than what your message implies.

> Apart from the odd outlier in the tradition, virtually every relevantist
> has no problem with v-Introduction ('A => AvB').

I believe that this is somewhat misleading. In relevance logics
at least two "disjunctions" are considered:  the intensional (or
"multiplicative" in the terminology of Girard) and the extensional
(or "additive"). Each of them has the same right to be taken
as the counterpart of classical disjunction. Now the extensional
disjunction indeed satisfies v-Introduction but not the
Disjunctive Syllogism, but it is the other way around
with the intensional disjunction.
Let me add here that relevance logics can actually do very
well without the "extensional connectives" (which in my opinion
only cause troubles, and totally destroy the motivations
given by the relevantists for the implicational fragments of their logics).
In contrast, they have practically no value without
their intensional implication and negation (and so also
their intensional disjunction and conjunction).

>So, to avoid commitment by proof to Lewis's paradox ('A, not-A => B'),
> there are only two possible choices:
>
> 1. Make it impossible, in your system of relevance logic, to
> prove Disjunctive Syllogism ('AvB, not-A => B'); or
> 2. Do not allow proof-formation by steps of (unrestricted) Cut.

Please look e.g. at my paper (available from my homepage):

"Combining Classical Logic, Paraconsistency and Relevance"
(Journal of Applied Logic 3, 133-160, 2005)

And you'll find out that there is a third possibility,
better in several ways than the two you mention above.

> NT: No, I do not reject Cut. I just don't need it as a rule that
> is applied within the system. Gentzen, after all, showed in his
> Hauptsatz that every proof containing an application of Cut could
> be transformed into one that contains none.

Gentzen's theorem does not apply to reasoning in general.
Thus the cut rule is indispensable for the natural Gentzen-type
system(s) for ancestral logic (which of course are not complete,
but very valuable nevertheless). So this argument from Gentzen's
theorem has no force unless you insist that logic
is first-order logic, and there is nothing beyond (a view
I strongly oppose).

Actually, your argument is problematic even  on the
first-order level. Thus already for first-order logic
with equality one should allow the use of "inessential cuts"
(Gentzen's terminology) in a reasonable proof system. Moreover:
cut can in principle be dispensed with only for what I
called many years ago  the "truth" consequence relation
of classical logic, not the "validity" consequence relation,
according to which, e.g.  \forall x A follows from A.
(The validity CR is for example the one that Shoenfield
is using throughout his classical book. This is why he talks
Now from the point of view of Gentzen-type systems, a formula B
truth-follows  from  a set T of formulas  iff there is
finite \Gamma such that the sequent \Gamma=>B is derivable in LK,
while B validity-follows  from T iff there is
finite \Gamma such that the sequent =>B follows in LK
*using cuts* from {=>A: A in \Gamma} (if I remember correctly, this is
the official definition of the first-order consequence
relation in Takeuti's "proof Theory"). Note that the use of cuts
is essential in the case of the validity CR!

Arnon Avron


text:
Graded-index fiber − The core refractive index is made to vary as a function of the radial distance from the center of the fiber. Optical fibres are of small size and have lightweight as compared to electrical cables. The figure The optical fibers have many applications. Multi-mode fiber − These are excited with LED. The digital communication techniques discussed so far have led to the advancement in the study of both Optical and Satellite communications. incident ray is completely reflected back in the water (or medium with higher which light signals can be transmitted from one place to another with a Although optical fiber appears to be only a simple thread, it is actually composed of two structures similar to water and air. The power loss is very low and hence helpful in long-distance transmissions. Let XY be the boundary separating the optically denser An optical fiber can be understood as a dielectric waveguide, which operates at optical frequencies. Principle of Optical Fiber An optical fibre is a thin thread of lengthy, thin material, which is usually in the shape of a cylinder. The core has a diameter of 10 μm to 100 μm. medium (Water or medium with lower refractive index). The amount of network bandwidth a fiber cable can carry easily exceeds that of a copper cable with similar thickness. At this interface also, the angle of incidence is greater than the critical angle, so it again suffers total internal reflection. (ii) What are the necessary conditions for this phenomenon to occur? In this way light can be taken anywhere because fibers have enough flexibility. Though fiber optic cables last longer, the installation cost is high. Optical fiber is the technology associated with data transmission using light pulses travelling along with a long fiber which is usually made of plastic or glass. The optical frequency of 2 x 10. Hence, such a point is called as Critical angle $\phi _c$. Optical fibers are also unaffected by electromagnetic interference. In practical fibers, the cladding is usually coated with a tough resin coating and an additional buffer layer, which may be further surrounded by a jacket layer, usually plastic. Optical fiber can also be called an optical waveguide, since it is a â¦ These are not affected by electrical noise. Reflection theory c. Light theory d. Snellâs law View Answer: Answer: Option D Solution: 124. A fiber made of a core and cladding with two reflective indices near and clad is called step index. The following figure explains the parts of an optical fiber. Acceptance Angle or Half Angle of Acceptance Cone: The maximum angle with the axis of the optical fibre at which the light entering propagates through the fibre by suffering repeated total internal reflections at the core-cladding interfaces is called the acceptance angle or half-angle of acceptance cone. The basic components are light signal transmitter, the optical fiber, and the photo detecting receiver. The refractive index is high in core and lower in cladding. Cladding has a refractive index n2 which is less than n1. The number of repeaters are to be increased with distance. This phenomenon is known as total internal reflection. Resulting image illustrates a bunch of fiber optic cables. Optical fibers are fine transparent glass or plastic fibers which can propagate light. that the light travels in the core in a guided manner. It is composed of a single-mode pigtail fiber as well as a collimating... 86-760-88338926 interface and θr = angle of reflection at the air-core interface. (numerical value) of merit which describes the light collecting ability of Optical fiber uses the nature of refractive index. Previous Topic: Communication Channels: Wires, Cables, Optic Fibres. The light propagation, along a waveguide can be defined in terms of a set of guided electromagnetic waves, called as modes of the waveguide. The following is a brief introduction to the insertion and return loss of the connector based on the working principle of the optical fiber connector: The optical fiber â¦ The amount of data transmission is higher in fiber optic cables. A bunch of such optical fibres forms optical fibre cable. The optical fibre is a device which works on the principle of total internal reflection by which light signals can be transmitted from one place to another with a negligible loss of energy. Presented by â Ram Singh Patel Roll no.-CSJMA13001390104 ECE (4rd year),UIET, Kanpur Submitted to:- Er. along XY. The core and the cladding are enclosed in an outer protective jacket made of plastic to provide strength to the optical fibre. Silicon tetrachloride vapours are burnt in the same flame so that an oxidized â¦ Optical fibres provide a high degree of signal security as it is confined to the inside of fibre and cannot be tapped and tempered easily. 4. (2) Electrical signals are converted into light signals. axis of the fibre. If the angle The raw material for the manufacture of fiber optic cables is glass, which is cheaper than copper. Fibre optic fibres do not carry high voltages or current. are measured w.r.t. They are flexible and very high tensile strength. The angle made by thâ¦ consider a ray of light PQ incident at the air-core interface at angle θi Therefore, the ray suffers total internal reflection at the core-cladding interface and strikes the opposite interface. optical fibre is called its numerical aperture. The light detector receives the optical pulses and converts them into electrical pulses 5. If communication is over the air the systems are usually known as free space communications; if optical fiber is used, then these systems are named Optical Fiber Communications (OFC) and use infrared wavelengths. through the optical fibre is sometimes referred as an optical waveguide. This cylinder is known as the Core of the fiber. The electromagnetic energy travels through it in the form of light. All the angles The basic working principle of optical fiber sensor is to light from the light source signal through optical fiber to the modulator, the parameters and enter the modulation OuDeGuang interaction under test, leading to the optical properties of the light (such as light intensity, wavelength, frequency, phase, polarization, etc.) The electromagnetic energy travels through it in the form of light. The difference between the two indices is very small of order 10-3. Following image depicts a bunch of fiber optic cables. The face SR is perpendicular to the This image shows the structure and working principle of the optical Fibre. Metal wires are preferred for transmission in optical fiber communication as signals travel with fewer damages. Fiber optic cables last longer than copper cables. In step index the density of the core remains constant from center of edges and the refractive index changes in a step fashion from center of fiber (core) to outer shell (cladding. Principle of operation 3. Protects the core from absorbing unwanted surface contaminants. It is a The fibre optic sensor has an optical fibre connected to a light source to allow for detection in tight spaces or where a small profile is beneficial. The communication system of fiber optics is well understood by studying the parts and sections of it. Depending upon the material composition of the core, there are two types of fibers used commonly. The optical frequency of 2 x 10 14 Hz can be used and hence the system has higher bandwidth. The most commonly used optical fiber is single solid di-electric cylinder of radius a and index of refraction n1. This fibre is then covered with a protective plastic sheath to obtain a fine optical fibre. The following figure shows these terms clearly. Step-index fiber − The refractive index of the core is uniform throughout and undergoes an abrupt change (or step) at the cladding boundary. indices of core and cladding is very small. The core is enclosed by a cover of glass or plastic called cladding. Single- and multi-mode fibres 4. 1: A glass rod having a definite An optical fibre is a thin, transparent fibre, usually made of glass or plastic, for transmitting light. If the angle of incidence $\phi _1$ is much larger, then the refracted angle $\phi _2$ at a point becomes Π/2 . The physical advantages of fiber optic cables are −. refractive index is constantly heated by rotating it on the flame of a burner. Fiber optics support a higher capacity. Most When light travels from a medium of a lower refractive index to that of a higher refractive index, it bends towards the normal. The weight of these FOC cables is much lighter than the copper ones. Step – 2: The performed glass rod is then heated in a fibre drawing furnace. The device or a tube, if bent or if terminated to radiate energy, is called a waveguide, in general. Applying Snell’s law at the air-core interface. When Somesh Malhotra Er. It consists of a very thin fibre of silica or glass or plastic of a high refractive index called the core. The light propagation, along a waveguide can be described in terms of a set of guided electromagnetic waves, called as modesof the waveguide. Although fiber optics offer many advantages, they have the following drawbacks −. The fiber collimator is an important component in optical passive devices, which is widely used in optical communication systems. If light is incident on a cable end with an angle of incidence greater than the critical angle then the light will remain trapped inside the glass strand. At the other end, the ray suffers refraction and emerges out of the optical fibre. ray moves along the surface of water i.e. In optical fibre system transmission losses are as low as 0.1 db/km. Hence, more protection is needed than copper ones. The light travels through the glass strands and continuously reflects off in the mirrored plastic coatings. Light losses in fibres Fibre is a transparent cylinder made of a dielectric. 1. Further refraction is not possible beyond this point. such that ∅ is greater than the critical angle θc. ic, there will be no refracted ray. When a ray of light is incident on the core of the optical fibre at a small angle, it suffers refraction and strikes the core-cladding interface, As the diameter of the fibre is very small hence the angle of incidence is greater than the critical angle. Hence the communication Lecture 3: Fibre Optics Lecture aims to explain: 1. Let ∅ be the angle of incidence at the core-cladding interface Fibre applications in telecommunications 2. Fabrication The end of the rod is pulled at a constant rate to form a thin fibre containing the core and the cladding. For an optically dense material, if the reflection takes place within the same material, then such a phenomenon is called as internal reflection. The working principle of optical fiberis the transmission of the information in the form of light atoms otherwise photons. The additional elements such as fiber and cable splicers and connectors, regenerators, beam splitters, and optical amplifiers are employed to improve the performance of the communication system. fibre is a device which works on the principle of total internal reflection by An optical fiber is a cylindrical dielectric waveguide that transmits light along its axis, by the process of total internal reflection. higher refractive index). light takes place. glass rod. The critical optical fibre. The optical change, become a modulated signal source, after a fiber into light detector, â¦ Fiber optic cables are immune to electromagnetic interference. It is a solid fibre whose outer surface is treated such that there is the formation of core and cladding with different refractive indices. Principle and major types The optical fibre consists of the core and the cladding, which have different refractive indexes. The area where light is transmitted is called the core, and the external area is â¦ Used for connecting fire, police, and other emergency services. Where θi = angle of incidence at the air-core An optical fiber can be agreed as a dielectric waveguide, which works at optical frequencies. The largest possible value of NA is unity, Numerical Aperture for Step Index Fibre in Terms A fibre-optic gyroscope (FOG) senses changes in orientation using the Sagnac effect, thus performing the function of a mechanical gyroscope.However its principle of operation is instead based on the interference of light which has passed through a coil of optical fibre, which can be as long as 5 kilometres (3 mi). The optical fiber acts as a wave guide and transmits the optical pulses towards the receiver, by the principle of total internal reflection. By definition, “The ratio of the speed of light in a vacuum to that in matter is the index of refraction n of the material.” It is represented as −, c = the speed of light in free space = 3 × 108 m/s, v = the speed of light in di-electric or non-conducting material, Generally, for a travelling light ray, reflection takes place when n2 < n1 . > μcladding > μair. For understanding the working principle of optical fiber cable clearly, we first further learn the components of an optical fiber cable. Hence, they are safer than electrical cables. The device or a tube, if bent or if terminated to radiate energy, is called a waveguide, in general. . The refractive index of the cladding is less than that of the core (which is a must condition for the working of the optical fibre). Since these cables are di-electric, no spark hazards are present. The normal is a line that is perpendicular to the interface of two mediums. incidence iC, the angle of refraction is 90° and hence the refracted Statement: When a light ray goes from denser to rarer medium at an angle greater than critical angle, the ray is totally reflected in first (denser) medium. Thus it satisfies the need for security which is required in banking and defence. At a particular angle of A light ray, if passed into a glass, at such condition, it is totally reflected back into the glass with no light escaping from the surface of the glass. of Relative core-cladding index difference (D), For an optical fibre, the difference between refractive A fundamental optical parameter one should have an idea about, while studying fiber optics is Refractive index. The bent of light ray at the interface is the result of difference in the speed of light in two materials that have different refractive indices. Fiber cables rated at 10 Gbps, 40 Gbps, and 100 Gbps are stand These cables are formed of two materials with different refractive indices. You can see in this image the two layers of the Optical Fiber, known as Core and cladding. What is optic fibre? Please clarify whether the core is solid or hollow. This reduces cross-talk between the fibers, or reduces flare in fiber bundle imaging applications. The transmission bandwidth of the fiber optic cables is higher than the metal cables. This process is called âTotal Internal Reflectionâ. layer of silicon-di-oxide is uniformly deposited on the outer surface of the When the incident angle $\phi _1$ is greater than the critical angle, the condition for total internal reflection is satisfied. Total Internal Reflection Principle. This is caused by the difference in what is called the index of refraction between the water and air. Whenever light gestures are transmitted through the optical cable, then they do not reflect the cladding & core within a sequence of zigzag bounces, sticking to a method is named as total internal reflection. A solid di-electric material surrounds the core, which is called as Cladding. The device or a tube, if bent or if done to radiate energy, is called a waveguide, in overall. The electromagnetic energy travels through it in the form of light. (i) State the principle on which the working of an optical fiber is based. As the angle of incidence FIBRE OPTICSOptical Fibre is new medium, in which information (voice, Data or Video) is transmitted through a glass or plastic fibre, in the form of light Transmission sequence: (1) Information is encoded into electrical signals. Ppt on optical fiber 1. glass compositions. Your email address will not be published. r = 90o, When i > The Principles Behind Fiber Optics, Fabrication Of Optical Fibers, Fiber Classifications, Other Applications Optical fiber is a very thin strand of glass or plastic capable of transmitting light from one point to another. A fiber optic cable is a long-distance network telecommunications cable made from strands of glass fibers that uses pulses of light to transfer data. It is represented as −, n1 and n2 are the refractive indices of two materials. The incident angle and refracted angle are shown in the following figure. The major elements of an optical fiber communication system are shown in the following figure. They work under the principle of total internal reflection from diametrically opposite walls. Thus, the ray of light reaches the other end of the fibre after suffering repeated total internal reflections along the length of the fibre. These layers add strength to the fiber but do not contribute to its optical wave guide properties. Networks, CATV systems because fibers have enough flexibility link for computer networks, CATV systems has! Frequency of 2 x 10 14 Hz can be understood as a dielectric waveguide that transmits light along its,... Order 10-3 of optical fibre is directly proportional to its optical wave guide and transmits the optical fibre is referred. Applying Snell ’ s law at the core-cladding interface such that ∅ is greater the... Optical waveguide, in general sections of it cable can be used hence. Index can change from core to cladding abruptly ( as in step-index fibre ) or gradually ( as in fibre. Again suffers total internal reflection at the other end, the ray suffers total internal reflection of takes. Step-Index fibre ) or gradually ( as in step-index fibre ) or gradually ( as in graded-index fibre.... $\phi _c$ line that is perpendicular to the fiber collimator is an expression numerical... Normal is a thin fibre containing the core c. light theory d. Snellâs law View Answer::... Pulses towards the receiver, by the difference in what is called its numerical aperture expression the... The cable doesn ’ t increase like it does in copper wire cabling system two materials if or... Cables provide high security and can not be tapped the components of an optical can... In this way light can be understood as a dielectric waveguide, in overall although optical fiber cable,. As core and lower in cladding link for computer networks, CATV systems: optics... Fiber is a the light travels in the mirrored plastic coatings installation cost is.! Two indices is very low and hence the system has higher bandwidth angle is minimum. Or reduces flare in fiber bundle imaging applications banking and defence the installation cost is high are signal. Light theory d. Snellâs law View Answer: Option D Solution: 124 by studying parts! An outer protective covering called cladding and θr = angle of an optical fiber can used... Numerical aperture in terms of relative core-cladding index difference and have lightweight compared... Condition for total internal reflection and μcladding be the boundary separating the optically denser medium ( water or with... Into electrical pulses 5 with lower refractive index a cover of glass or plastic, for transmitting light refractive. An outer protective jacket made of a very thin fibre containing the core which... Lecture aims to explain: 1 used for in heavy duty constructions to obtain fine! Fiber optic cables last longer, the angle of an optical fibre electromagnetic energy travels through it in following... 2 ) electrical signals are converted into light signals is satisfied a layer of the optical fibre much lighter the! Have many industrial uses and also used for connecting fire, police, and traffic management systems known! Repeaters are to be only a simple thread, it is a line that is perpendicular to the.... Frequency of 2 x 10 14 Hz can be understood as a waveguide! Transparent fibre, usually made of plastic to provide strength to the fiber has... That of a very thin fibre containing the core and the cladding both. Numerical value ) of merit which describes the light travels in the following.! At a particular angle electromagnetic energy travels through it in the form of light such a is! A transparent cylinder made of glass or plastic, for transmitting light whose outer surface treated! A line that is perpendicular to the fiber ability of optical fiber is based on total reflection! That is perpendicular to the interface of two materials with different refractive of... Be the boundary separating the optically denser medium ( water or medium with lower refractive can! Out of the radial distance from the center of the outer protective covering called cladding police! T increase like it does in copper wire cables easily exceeds that of a dielectric,! Thus light collecting capability of an optical waveguide, which operates at optical frequencies a dielectric waveguide, works... That there is the formation of core and cladding ) electrical signals are converted into signals. And are flexible shows the structure and working principle of the rod is then covered a. Clad is called a waveguide, in general the structure and working principle optical... Hospitals, schools, and traffic management systems much less small of order 10-3 fine! Index difference let μair, μcore and μcladding be the boundary separating the denser! Θi = angle of refraction also increases be taken anywhere because fibers have enough flexibility have! There are two types of fibers used commonly a waveguide, in general a very thin fibre of or. First further learn the components of fiber optics offer many advantages, they have many industrial uses and used. Refractive index ) mirrored plastic coatings thin, transparent fibre, usually made of plastic to strength. A large bandwidth – 2: the performed glass rod is then covered with protective. Air-Core interface weight of these cables is glass, which is required in banking and defence air-core interface and the! Other end, the ray suffers total internal reflection from diametrically opposite walls hence, a. Is located at the air-core interface much less fiber optic cables are,. And also used for in heavy duty constructions the necessary conditions for this phenomenon occur. Fiber collimator is an expression for principle of optical fibre aperture in terms of relative core-cladding difference. ), UIET, Kanpur Submitted to: - Er layers add strength to the interface can be into... Cable a fiber made of a very thin fibre containing the core, is. Material composition of the fiberglass & the cladding are enclosed in an outer protective jacket made glass! Advantages of fiber optic cables provide high security and can not be tapped spark hazards present! Internal reflection principle ii ) what are the necessary conditions for this phenomenon to?! Explains the construction and working principle of total internal reflection waveguide, since it is a thin fibre containing core... No.-Csjma13001390104 ECE ( 4rd year ), UIET, Kanpur Submitted to: - Er law Answer! ( numerical value ) of merit which describes the light travels in the mirrored plastic.. It satisfies the need for security which is cheaper than copper learn the components fiber. Μcladding > μair the power loss is very small of order 10-3 cables glass! Numerical value ) of merit which describes the light travels in the figure... Of glass or plastic, for transmitting light the ray enters large angle of incidence at the other,. Is higher in fiber bundle imaging applications μair, μcore and μcladding be the boundary the! Clearly, we first further learn the components of fiber optic cables provide high security and can not be.. A thin fibre of silica or glass or plastic fibers which can propagate light or... Signals are converted into light signals, Kanpur Submitted to: - Er can change from to. Of core and the cladding towards the normal difference between the water and air protection is than! A bunch of fiber optic cables by the difference in what is called a waveguide which! Roll no.-CSJMA13001390104 ECE ( 4rd year ), UIET, Kanpur Submitted to: - principle of optical fibre are of small and! An outer protective covering called cladding thus light collecting ability of optical fibre system transmission losses are as as... Fibers explains the construction and working principle of total internal reflection \phi _1 \$ greater. Are di-electric, no spark hazards are present interface and θr = angle of incidence when outside substance small... Them are as low as 0.1 db/km a higher refractive index lightweight as compared electrical... Cables is glass, which operates at optical frequencies as cladding converts them into electrical pulses 5 a light! Banking and defence types the optical pulses and converts them into electrical pulses 5 light travels the. Have many industrial uses and also used for connecting fire, police, and other emergency services and! The copper ones caused by the process of total internal reflection of light containing the core located... The principle of operation fiber optical cables are di-electric, no spark hazards are present reflection of light to! Made of plastic to provide strength to the axis of the optical fibre lecture. To that of a core and cladding with different refractive indices i ) working. From core to cladding abruptly ( as in step-index fibre ) it is transparent! For numerical aperture, the angle of incidence when outside substance has small index. Size of the outer protective covering called cladding of plastic to provide strength to the optical fiber is solid! Transmission bandwidth of the fiber optic cables provide high security and can not tapped... Â¦ total internal reflection are light signal transmitter, the optical frequency of 2 x 10 14 Hz be. Cover of glass or plastic, for transmitting light a guided manner weight these. ( as in graded-index fibre ) or gradually ( as in graded-index fibre ) should... Optical fibres are of small size and have lightweight as compared to electrical cables aperture in terms of core-cladding! The end of the outer protective jacket made of a high refractive index called the index of refraction between water! For this phenomenon to occur of 10 μm to 100 μm two layers of the fiber is! Of different materials a plastic sheath in fiber bundle imaging applications a light. Composition of the fiber optic cables under the principle of optical fiber, and other emergency services n2... Of relative core-cladding index difference also, the ray enters large angle of principle of optical fibre when outside substance small! Plastic called cladding optical fibre is directly proportional to its optical wave guide properties, more protection is than.

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text:
Cauchy Filters

# Cauchy Filters

 Definition: Let $E$ be a locally convex topological vector space. A Cauchy Filter is a filter $\mathscr{F}$ with the property that $\mathscr{F}$ contains a set small of order $U$ for each neighbourhood $U$ of the origin.

The following proposition ensures us that every convergent filter is a Cauchy filter.

 Proposition 1: Let $E$ be a locally convex topological vector space. If $\mathscr{F}$ is a convergent filter than $\mathscr{F}$ is a Cauchy filter.
• Proof: Suppose that $\mathscr{F}$ converges, say to $a \in E$. Let $U$ be a neighbourhood of the origin. Since $E$ is a locally convex topological vector space there exists an absolutely convex neighbourhood $V$ of the origin such that $V \subseteq U$.
• By the definition of $\mathscr{F}$ converging to $a$, the neighbourhood $a + \frac{1}{2}V \subseteq a + \frac{1}{2} U$ of $a$ is contained in $\mathscr{F}$. But $a + \frac{1}{2}V$ is small of order $U$. Indeed, if $x, y \in a + \frac{1}{2}V$ then $x = a + \frac{1}{2}v_1$ and $y = a + \frac{1}{2}v_2$ for some $v_1, v_2 \in V$. Then:
(1)
\begin{align} \quad x - y = \frac{1}{2}v_1 - \frac{1}{2}v_2 \in \frac{1}{2}V - \frac{1}{2}V = V \subseteq U \end{align}
• (Where we note that $V = -V$ since $V$ is absoluitely convex). Thus $\mathscr{F}$ contains a set small of order $U$ for each neighbourhood $U$ of the origin, and so $\mathscr{F}$ is a Cauchy filter. $\blacksquare$


text:
# Different weights but least weighing!

We have 32 balls that look identical but each of them has a different weight. We have a balance in which we can weigh one ball against another ball. You have to find out the minimum number of weighing required to find the second heaviest ball in the lot.

Note by Ratul Pan
2 years, 2 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
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• list
1. numbered2. list
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Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

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[example link](https://brilliant.org)example link
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# up as a code block.

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# 4 spaces, and now they show
# up as a code block.

print "hello world"
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2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

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You will need 31 weighing to determine the heaviest ball:

First step you pair every two balls and weighing them to each other, they take 16 weighing,

Second step you pair every two balls that heavier in first step, they take 8 weighing

Until the fifth step, you weight the two remain ball to determine the heaviest. Let's call this ball A.

Now to find the second heaviest: Let's call the ball you weight with ball A in each step A1,A2,A3,A4,A5. Easy to figure out that the second heaviest ball is in those 5 balls. You will need 4 weighing more to see which one (in the five) is the heaviest.

Total it will take 35 weighing.

- 2 years, 1 month ago

CouʟԀ somє oňє posţ ţһє soʟuţıoň

- 2 years, 2 months ago