text:
## Functions whose Inverses are Functions

Standard post by ancoop on November 12, 2018
Comments are off for this post

Recall that we defined inverses for any relation:

Definition. If $R$ is a relation from $A$ to $B$, then $R^{-1}$ is a relation from $B$ to $A$, defined by $(b,a)\in R^{-1}$ iff $(a,b)\in R$.

Thus (in our formalism) every function has an inverse. The inverse of a function so defined, however, may not be a function itself! (Remember that most operations we do to sets or to relations will not preserve functionhood.) So let’s discover when taking the inverse (“swapping $x$ and $y$”) actually does result in a function.

Recall that $g:A\rightarrow B$ means that two conditions hold:

1. $\operatorname{Domain}(g)=A$
2. $\left((a,b)\in g\text{ and }(a,c)\in g\right)\Rightarrow b=c$.

Exercise 1. Suppose $f:X\rightarrow Y$. By substituting $f^{-1}$ for $g$, $X$ for $A$, and $Y$ for $B$, write what it means for $f^{-1}:Y\rightarrow X$.

Exercise 2. Use the definition of $f^{-1}$ to rewrite your conditions solely in terms of $f$ (with no $f^{-1}$ anywhere).

Definition.  Let $f:X\rightarrow Y$. We call $f$ onto or surjective if $\forall y\in Y, \exists x\in X: y=f(x)$. We call $f$ one-to-one or injective if for any $x,z\in X$, we have that $f(x)=f(z)$ guarantees $x=z$.

Definition. A function which is both one-to-one and onto is called a one-to-one correspondence or a bijection.

The following theorem is important, but its proof is just the combination of Exercises 1 and 2.

Theorem. Let $f:X\rightarrow Y$. Then $f^{-1}:Y\rightarrow X$ iff $f$ is both one-to-one and onto.

Theorem.  Suppose $f:X\rightarrow Y$ is one-to-one and $g:\operatorname{Range}(f)\rightarrow Z$ is one-to-one. Then $g\circ f:X\rightarrow Z$ is one-to-one.

Proof. Note that the condition that $g$ be defined on the range of $f$ is necessary in order to compose the two functions as functions.

Let $x,z\in X$. Suppose $g\circ f(x)=g\circ f(z)$. Applying the fact that the composition of functions is a function, we have \begin{align*}g(f(x))=g(f(z))\end{align*}

Since $g$ is one-to-one, we have $f(x)=f(z)$. Since $f$ is one-to-one, we have $x=z$. Since $x,z$ were arbitrary, we’ve shown $g\circ f$ is one-to-one. $\Box$.

Theorem. Suppose $f:X\rightarrow Y$ is onto and $g:Y\rightarrow Z$ is onto. Then $g\circ f$ is onto.

Proof. This is an exercise for you.

## Sets of Functions

Standard post by ancoop on November 11, 2018
Comments are off for this post

Consider the set of all functions $f:X\rightarrow \{0,1\}$. Each subset $A\subseteq X$ has such a function, namely its characteristic function, $\chi_A$; moreover, given $f:X\rightarrow\{0,1\}$ we can define the set $A=f^*(\{1\})$ and see that $f=\chi_A$. So the functions $f:X\rightarrow \{0,1\}$ correspond to subsets of $X$.

Definition. Given sets $A$, $B$, define \begin{equation*}A^B=\left\{f:B\rightarrow A\right\}\end{equation*} to be the set of all functions from $B$ to $A$.

This notation connects with our powerset notation by writing $2$ for the set $\{0,1\}$ (which has two elements).

Theorem. If $A$ has exactly $n$ elements and $B$ has exactly $m$ elements, then $A^B$ has exactly $n^m$ elements.

Proof. To define a function $B\rightarrow A$ requires choosing, for each $b\in B$, one of the $n$ elements of $A$.

Each time we make such a choice, there are $n$ options; and we have to make the choice $m$ times. So in total there are $n\cdot n\cdot\cdots \cdot n=n^m$ possibilities. $\Box$

## Functions and Sets

Standard post by ancoop on November 11, 2018
Comments are off for this post

[callout headingicon=”noicon” textalign=”textleft” type=”basic”]All in all, those who were chained would consider nothing besides the shadows of the artifacts.

Plato, The Allegory of the Cave

[/callout]

### Pushforwards and Pullbacks

A function $f:X\rightarrow Y$ gives us a way of taking elements in $X$ and pushing them over to $Y$. We call $f(x)$ the image of $x$ under $f$. In fact, we can also think of $f$ as giving a way to push a set of elements of $X$ over to $Y$.

Definition. Given $f:X\rightarrow Y$ and $A\subseteq X$, we define $f_*A=\left\{f(x)\middle\vert x\in A\right\}$. We call $f_*A$ the image of $A$ under $f$ or the pushforward of $A$ by $f$.

Notice that by definition, $f_*A$ is a subset of the target $Y$.

E.g. If $f:t\mapsto t^2$, and $A=[-2,3]$, then $f_*A=[0,9]$.

Proof.  ($\Rightarrow$) Let $y\in f_*A$. Then $\exists x\in A: f(x)=y$. So $-2\leq x\leq 3$.

If $-2\leq x\leq 0$, then $y=x^2$ is between $0$ and $4$. So $y\in[0,9]$.

If $0\leq x\leq 3$, then $y=x^2$ is between $0$ and $9$. So $y\in[0,9]$.

($\Leftarrow$) Let $y\in[0,9]$. Then $0\leq y\leq 9$. Let $x=\sqrt{y}$. So $0\leq x\leq 3$. Then $f(x)=\left(\sqrt{y}\right)^2=y$. Since $0\leq x\leq 3$, $x\in[-2,3]$.

Notation. Most sources write $f(A)$ for the image of $A$ under $f$. This is not an entirely unreasonable choice, but it means that the symbol $f(A)$ has two interpretations — if $A$ is an element of $X$, then $f(A)$ is an element of $Y$; if $A$ is a subset of $X$, then $f(A)$ is a subset of $Y$. Our notation $f_*A$ is always going to be a set. But you should know, for your future mathematics courses, that $f(A)$ is a standard notation.

Theorem/Realization. If $f:X\rightarrow Y$, then $f_*:2^X\rightarrow 2^Y$.

That is, $f_*$ takes subsets of $X$ and yields subsets of $Y$.

Definition. Given $f:X\rightarrow Y$ and $B\subseteq Y$, we define $f^*B=\left\{x\middle\vert f(x)\in B\right\}$. We call $f^*B$ the preimage of $B$ under $f$ or the pullback of $B$ by $f$.

Note that $f^*B$ is a subset of the domain $X$.

E.g. If $f:\mathbb{R}\rightarrow\mathbb{R}$ is given by $t\mapsto t^2$, then $f^*((-2,-1))=\varnothing$ and $f^*([-4,4))=[0,2)$.

Theorem/Realization. If $f:X\rightarrow Y$, then $f^*:2^Y\rightarrow 2^X$.

Notation. Just like with the image, there is a standard-but-terrible notation for the preimage. Most sources write $f^{-1}(B)$ for the preimage of $B$ under $f$. This is just nutty, because the object in question doesn’t have anything to do with inverses. Nevertheless, you should expect at least once in your mathematical life to have to deal with the string of symbols $f^{-1}(B)$, where $B$ is some set. We live in a fallen world.

Theorem. Let $f:X\rightarrow Y$, $A\subseteq X$, $B\subseteq X$, $C\subseteq Y$, $D\subseteq Y$. Then

1. $f_*(A\cup B)=f_*(A)\cup f_*(B)$
2. $f_*(A\cap B)\subseteq f_*(A)\cap f_*(B)$
3. $f^*(C\cup D)=f^*(C)\cup f^*(D)$
4. $f^*(C\cap D)=f^*(C)\cap f^*(D)$
5. $f^*(C^c)=\left(f^*(C)\right)^c$

Proof. This is an exercise, but it’s just an exercise in proving set equalities.

### Characteristic Functions

Recall that for $S$ to be a set means that “$x\in S$” is an open sentence. We can view this in terms of functions.

Definition. Given a set $A$, the characteristic function of $A$ is the function \begin{equation*} \chi_A(x)=\begin{cases}1 &\text{ if } x\in A\\0&\text{ otherwise}\end{cases} \end{equation*}

Technically in order to define this function, we need to specify a universe for $x$ — but notice that it really doesn’t much matter, because as long as $x\notin A$, $\chi_A(x)=0$.

Exercise. If $A$ and $B$ are subsets of some common universe $U$, then for any $x\in U$, we have:

1. $\chi_{A\cap B}(x)=\chi_A(x)\chi_B(x)$,
2. $\chi_{A\cup B}(x)=\max\{\chi_A(x),\chi_B(x)\}$
3. $\chi_{A^c}(x)=1-\chi_{A}(x)$.

Theorem.  $\chi_A^*(\{1\})=A$.

This says that $A$ determines its characteristic function, and $\chi_A$ determines $A$ as well.

## Operations on Functions

Standard post by ancoop on November 10, 2018
Comments are off for this post

### Set Operations on Functions

Functions are sets, so we can try to combine them like we combine sets. That is, we can ask the following questions:

• Is the complement of a function necessarily a function?
• Is the intersection of two functions necessarily a function?
• Is the union of two functions necessarily a function?
• Is the set product of two functions necessarily a function?

Let’s investigate the first three; the last one I’ll leave for homework.

To start with, let’s pick two functions and run with them. How about: $f:x\mapsto x+1$ and $g:t\mapsto t^2$.

The complement of $g$ is $g^c=\left\{(x,y)\middle\vert y\neq x^2\right\}$, shown here in hatching:That’s definitely not a function: it badly fails the vertical line test.

What about intersection?

The intersection is just two points; so $f\cap g$ is not a function from $\mathbb{R}$ to $\mathbb{R}$. But $f\cap g$ is a function from a smaller set.

Let’s look at $f\cup g$:

It also fails the vertical line test. But:

The Pasting Lemma (in the category of sets). Let $f:X\rightarrow Y$ and $g:A\rightarrow B$ be two functions. Then $f\cup g:X\cup A\rightarrow Y\cup B$ if and only if $f|_{A\cap X}=g|_{A\cap X}$.

Proof. This is a biconditional, so we have two directions:

($\Leftarrow$) Assume $f|_{A\cap X}=g|_{A\cap X}$. We need to show $f\cup g$ is a function; to do this, there are two clauses.

(clause 1: domain) Let $t\in X\cup A$. Then either $t\in X$ or $t\in A$.

If $t\in X$, then because $f:X\rightarrow Y$, $\exists y\in Y: (t,y)\in f$. Then $(t,y)\in f\cup g$.

If $t\in A$, then because $g:A\rightarrow B$, $\exists b\in Y: (t,b)\in g$. Then $(t,b)\in f\cup g$.

In either case, there’s an element of $z\in Y\cup B$ so that $(t,z)\in f\cup g$. That’s clause 1 of the definition of function.

(clause 2: unambiguousness) Let $t\in X\cup A$ and $z_1,z_2\in Y\cup B$ be such that $(t,z_1)\in f\cup g$ and $(t,z_2)\in f\cup g$. There are four cases:

If $(t,z_1)\in f$ and $(t,z_2)\in f$, then the fact that $f$ is a function implies $z_1=z_2$.

If $(t,z_1)\in g$ and $(t,z_2)\in g$, then the fact that $g$ is a function implies $z_1=z_2$.

If $(t,z_1)\in f$ and $(t,z_2)\in g$, then we have $t\in \operatorname{Domain}(f)=X$ and $t\in\operatorname{Domain}(g)=A$, so $t\in X\cap A$. But then $z_1=f(t)=f|_{X\cap A}(t)=g|_{X\cap A}(t)$. So $(t,z_1)\in g|_{X\cap A}\subseteq g$. So $(t,z_2)\in g$. Since $g$ is a function, $z_1=z_2$.

If $(t,z_1)\in g$ and $(t,z_2)\in f$, then we have $t\in \operatorname{Domain}(f)=X$ and $t\in\operatorname{Domain}(g)=A$, so $t\in X\cap A$. But then $z_2=f(t)=f|_{X\cap A}(t)=g|_{X\cap A}(t)$. So $(t,z_2)\in g|_{X\cap A}\subseteq g$. So $(t,z_1)\in g$. Since $g$ is a function, $z_1=z_2$.

In any of these four cases, $z_1=z_2$, which is what we needed for clause 2 of the definition of function.

($\Rightarrow$) In this direction, we need to show two functions are equal. So we should probably use the TAWFAE. I’ll leave this direction to you.

Why does this theorem bear such a silly name? Consider this diagram, which tells you how to build a cube from a piece of paper. In order to attach the sides, you have to paste the red tab to the back of one of the appropriate square (I’ve indicated where the red tab gets attached by coloring it green). We won’t succeed in pasting unless the red tab and the green target match up.

That’s what the condition $f|_{X\cap A}=g|_{X\cap A}$ ensures: that the functions $f$ and $g$ agree on the overlap of their domains.

Again, an example from precalculus mathematics: when we define a function piecewise, like $$h(x)=\begin{cases}x^2 & \text{ if } x\leq 3\\ -x+1&\text{ if } x>3\end{cases}$$we usually want to make sure that the two pieces have nonoverlapping domains. This corresponds, in the Theorem above, to the case that $X\cap A=\varnothing$. If we weren’t so careful, we might have a situation like this:$$h(x)=\begin{cases}x^2 & \text{ if } x\leq 3\\ -x+1&\text{ if } x>1\end{cases}$$where it’s ambiguous what $h(2)$ is:

### Composition of Functions

Because functions are relations, we can consider their composites. We have the following nice theorem:

The Composition of Functions Is a Function. If $f:X\rightarrow Y$ and $g:Y\rightarrow Z$, then $g\circ f:X\rightarrow Z$.

Proof. There are two things to show:

(clause 1: domain) Let $x\in X$. Then, because $f$ is a function, $\exists y\in Y:(x,y)\in f$. Since $y\in Y$ and $g$ is a function, $\exists z\in Z:(y,z)\in g$. Then $(x,z)\in g\circ f$. $\Box$.

(clause 2: unambiguousness) Given $x\in X, z_1,_2\in Z$ so that $(x,z_1)\in g\circ f$ and $(x,z_2)\in g\circ f$. We need to show $z_1=z_2$.

Since $(x,z_1)\in g\circ f$, there is some $y_1\in Y$ with $(x,y_1)\in f$ and $(y_1,z_1)\in g$. We also have $y_2\in Y$ with $(x,y_2)\in f$ and $(y_2,z_2)\in g$.

Since $f$ is a function, $y_1=y_2$. So we have $(y_1,z_2)\in g$. But since $g$ is a function, $(y_1,z_2)\in g$ and $(y_1,z_1)\in g$ forces $z_1=z_2$. $\Box$.

Important Formula. If $f$ and $g$ are functions, so that $g\circ f$ is a function, then $(g\circ f)(x)=g(f(x))$.

Proof. What does it mean for $z=(g\circ f)(x)$? It means $(x,z)\in g\circ f$. But that means $\exists y: (x,y)\in f$ and $(y,z)\in g$. So $y=f(x)$ and $z=g(y)$. By substituting, we obtain $z=g(f(x))$. $\Box$

This formula explains the “backwards” looking choice of how we write compositions: even though $f$ come first, we write $g\circ f$. The only way out of this would be to switch our function notation to something like $(x)f$, so that the composite would be $((x)f)g$. (Some people do this, and it has a name: reverse Polish notation. But most normal people don’t do it.)

## Domains and Rules

Standard post by ancoop on November 10, 2018
Comments are off for this post

When are two functions equal? A function is a kind of relation, which means it’s a kind of set. So two functions being equal means, fundamentally, that each is a subset of the other. That’s inconvenient, and it doesn’t really jibe with our understanding of functions as encoding rules.

Theorem About When Functions Are Equal (TAWFAE). Let $f$ and $g$ be functions. Then $f=g$ if and only if:

1. $\operatorname{Domain}(f)=\operatorname{Domain}(g)$, and
2. $\forall x\in \operatorname{Domain}(f), f(x)=g(x)$.

Proof. This theorem sort of looks like it will be complicated to prove, but let’s keep our wits about us and rely on logic to help us out.

Globally, this is a biconditional proof so it has two directions:

($\Rightarrow$) Assume $f=g$. We need to prove two things:

(clause 1) This is the claim that two sets are equal, so we have to prove each is a subset of the other:

($\subseteq$) Let $t\in \operatorname{Domain}(f)$. Then $\exists y: (t,y)\in f$. Since $f=g$, this means $(t,y)\in g$. So $t\in\operatorname{Domain}(g)$.

($\supseteq$) Let $t\in \operatorname{Domain}(g)$. Then $\exists y: (t,y)\in g$. Since $f=g$, this means $(t,y)\in f$. So $t\in\operatorname{Domain}(f)$.

(clause 2) This is a universal claim, so we start the same way as always: Let $x\in \operatorname{Domain}(f)$. Then $\exists y: (x,y)\in f$. We can rewrite this as $y=f(x)$. On the other hand, we know from clause 1 that $\operatorname{Domain}(g)=\operatorname{Domain}(f)$, so $\exists z: (x,z)\in g$. We can rewrite this as $z=g(x)$. But since $g=f$, we actually have $(x,z)\in f$.

Since $(x,y)\in f$ and $(x,z)\in f$, and $f$ is a function, $y=z$.

So $f(x)=y=z=g(x)$.

($\Leftarrow$) Assume both clauses 1 and 2 hold. In this direction, we need to prove two sets are equal.

($\subseteq$) Let $Q\in f$. Then $Q=(x,y)$ for some $x$ and some $y$. $x\in\operatorname{Domain}(f)$, so by clause 1, $x\in\operatorname{Domain}(g)$. Then there’s some $z$ with $(x,z)\in g$. We can rewrite in function notation as $y=f(x)$ and $z=g(x)$. By clause 2, we know $f(x)=g(x)$, so $y=z$. That is, $Q=(x,y)\in g$.

($\supseteq$) This is so similar to $\subseteq$, I’ll let you work it out.

Since we proved both directions, we’re done. $\Box$.

This theorem gives us a new way to prove two functions are equal, should we ever need it: we establish clause 1 (“they have the same domain”) and clause 2 (“they have the same rule”).

### Restrictions and Extensions

In middle and high school algebra courses, we often ask questions like “What’s the domain of $f(x)=\frac{sin(x)}{x^2-25}$?”. According to the TAWFAE, this is actually a dumb question: TAWFAE says in order to know what a function is, we need to know what the domain is. So a question like doesn’t actually contain enough information to be answerable.

What the question is really asking, though is: what’s the most natural, or maybe what’s the largest possible domain on which this rule makes sense?

As a middle-school student, I was very confused by questions like this one:

Find the domain of $f(x)=\frac{x^2-9}{x+3}$.

My answer would usually go like this: that function is really the same as $x-3$, the domain of which is all of $\mathbb{R}$. And I would always get that question wrong. Why?

The functions $\psi:t\mapsto \frac{t^2-9}{t+3}$ and $\varphi:x\mapsto x-3$ do indeed have the same rule, in the sense that, whenever $\psi(t)$ makes sense, we have

$$\psi(t)=\frac{t^2-9}{t+3}=\frac{(t-3)(t+3)}{t+3}=t-3=\varphi(t)$$

But the natural domain of $\psi$ is $\mathbb{R}\setminus\{-3\}$ and the natural domain for $\varphi$ is $\mathbb{R}$; so that means the domains are different — hence by the TAWFAE, we know $\psi\neq\varphi$. Nevertheless, the two functions are related.

Definition. If $f:X\rightarrow Y$ and $A\subseteq X$, we define the restriction of $f$ to $A$ as

\begin{align*}f|_{A}:A&\rightarrow Y\\ a&\mapsto f(a)\end{align*}

That is, $f|_A$ has the same rule as $f$, just we only care about inputs that come from $A$.

Equipped with this notion, we can describe the relationship between $\psi$ and $\varphi$:

$$\psi=\varphi|_{\mathbb{R}\setminus\{-3\}}$$

Definition. If $f:X\rightarrow Y$, and $X\subseteq Z$, and there is $g:Z\rightarrow Y$ with $g|_X=f$, we call $g$ an extension of $f$.

Here’s an illustration: if we let $f$ denote the function displayed in green, $g$ denote the function displayed in blue, and $h$ the function displayed in red, then we have that:

• $f|_{[-2,\infty)}=g$
• $f|_{[3,\infty)}=h$
• $g|_{[3,\infty)}=h$
• $f$ is an extension of $g$, from $[-2,\infty)$ to all of $\mathbb{R}$
• $g$ is an extension of $h$, from $[3,\infty)$ to $[2,\infty)$

## The Basics of Functions

Standard post by ancoop on November 9, 2018
Comments are off for this post

Definition. A function $f$ from a set $X$ to a set $Y$ is a relation which satisfies the following two conditions:

1. $\forall x\in X, \exists y\in Y: (x,y)\in f$, and
2. $\forall x\in X, y_1,y_2\in Y, \left[(x,y_1)\in f \wedge (x,y_2)\right]\in f\Rightarrow y_1=y_2$

That is, a function assigns to each (clause 1) element of $X$ an element of $Y$, and that assignment is unambiguous (clause 2). Since we could rewrite clause 1 as $\operatorname{Domain}(f)=X$ (why?), we’ll refer to clause 1 as the “domain clause” and clause 2 as the “unambiguousness clause”.

Notation. Instead of the mouthful $f$ is a function from $X$ to $Y$, we write the shorthand $f:X\rightarrow Y$.

We also have this familiar notation:

Notation. If $f$ is a function from $X$ to $Y$, and $(x,y)\in f$, we write $y=f(x)$.

Observe that if $(x,y_1)\in f$ and $(x,y_2)\in f$, then this notation reads as follows: $y_1=f(x)$ and $y_2=f(x)$, so we’d darn well better have $y_1=y_2$, or else something is very wrong.

There is another way to represent a function, by specifying its domain and its rule:

Notation. If $f$ is a function from $X$ to $Y$, we write $f:X\rightarrow Y$. To specify the rule that $f$ uses, we write \begin{align*}f:X&\rightarrow Y\\x&\mapsto f(x) \end{align*}

For example, we could denote the function which takes in a real number as input, and squares it by

\begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ t&\mapsto t^2 \end{align*}

Notice that the arrow between the sets is a different shape from the one between elements of the set. As usual we want to let go of our $x$s, so we could as well have written

\begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ z&\mapsto z^2 \end{align*}

or

\begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ Q&\mapsto Q^2 \end{align*}

This frees us from the tyranny of always writing $y=f(x)$.

## Functions

Standard post by ancoop on November 2, 2018
Comments are off for this post

[callout headingicon=”noicon” textalign=”textleft” type=”basic”]On the road to the City of Skepticism, I had to pass through the Valley of Ambiguity.

Adam Smith

[/callout]

What is a function? The notion of a function is fundamental to mathematics, but like many fundamental aspects of mathematics, its definition was not nailed down until relatively recently. Mathematicians were using functions without thinking all that hard about what functions were! Let’s rectify that.

The definition that I prefer to use is the following:

Definition. function from the set $X$ to the set $Y$ is an unambiguous rule which assigns to each element of $X$ an element of $Y$.

By unambiguous, I mean that each $x\in X$ has only one element of $Y$ assigned to it. This may seem obvious, but in fact there are some interesting mathematical operations that turn out to be a little ambiguous, for example:

Definition. We say that $t$ is a square root of $x$ if $t^2=x$.

It’s not hard to see that both $-7$ and $7$ are square roots of $49$, so the phrase “the square root of $49$” is actually ambiguous. There are various ways to deal with such ambiguity: for example, we could just content ourselves with the symbol $\pm 7$, representing the two-element set $\{-7,7\}$, but then the square root of a number isn’t a number!. The most straightforward way is just to exclude it by assumption, as I’ve done with the definition of function I gave above.

To functionize the notion of square root, we usually do the following:

Definition. We say that $t$ is the principal square root of $x$, and write $t=\sqrt{x}$, if $t^2=x$ and $t\geq 0$.

That is, if there are the two possible choices for square root, we decide always to take the positive one. This notion of “the square root” is now unambiguous, i.e. a function.

Let’s connect this idea back to our formalism with relations. It’s clear that $t^2=x$ expresses a relationship between $t$ and $x$, so we can hope that our work on relations will help us develop a good definition of function as a kind of relation. For a function from $X$ to $Y$, we will want to consider pairs $(x,y)$, so our relation (let’s be traditional and call it $f$) will be a subset of $X\times Y$.

What would it mean for the rule to be ambiguous? There would be some $x\in X$ with $(x,y_1)\in f$ and $(x,y_2)\in f$, where $y_1$ and $y_2$ are distinct. That is, $f$ being ambiguous means

\begin{equation*}
\exists x\in X,\exists y_1\in Y, \exists y_2\in Y: (x,y_1)\in f \wedge (x,y_2)\in f\wedge y_1\neq y_2
\end{equation*}

Activity. Write a denial of the above logical sentence. This is what it means for $f$ to be unambiguous.

Activity. If your denial only includes $\sim$ and $\vee$, alter it using logical facts so that it contains a $\Rightarrow$.


text:
THE ETHYNYL RADICAL (CCH) DETECTED BY LASER MAGNETIC RESONANCE

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/10360

Files Size Format View
1978-FB-08.jpg 101.2Kb JPEG image

 Title: THE ETHYNYL RADICAL (CCH) DETECTED BY LASER MAGNETIC RESONANCE Creators: Saykally, R. J.; Evenson, K. M. Issue Date: 1978 Publisher: Ohio State University Abstract: We have detected absorptions at 490 $\mu m$ by a new” transient paramagnetic molecule generated in several hydrocarbon-fluorine atom flames in a $CO_{2}$-pumped far-infrared laser magnetic resonance spectrometer. The new species has been tentatively established through isotopic substitution experiments to be the $\sigma$-radical CCH $(X^{2}\Sigma^{-})$- previously detected $^{1}$ in interstellar gas clouds and in rare gas matrices, but never before observed terrestrially in the gas phase. The strongest CCH absorptions have been produced in a $CH_{4}$ -F flame under conditions which also yield intense magnetic resonance spectra from CH and $CH_{2}$. Additional transitions are being sought and analysis of the 490 $\mu m$ spectrum is underway. Description: $^{1}$K. D. Tucker, K. L. Kutner, and P. Thaddeus, Astrophys. J. 193, L115 (1974)."" Author Institution: National Bureau of Standards URI: http://hdl.handle.net/1811/10360 Other Identifiers: 1978-FB-08

format: plain


text:
Next: 3.2 Structure of Relaxed Up: 3. Strain Effects on Previous: 3. Strain Effects on

Subsections

# 3.1 Theory of Elasticity

The property of solid materials to deform under the application of an external force and to regain their original shape after the force is removed is referred to as its elasticity. The external force applied on a specified area is known as stress, while the amount of deformation is called the strain. In this section, the theory of stress, strain and their interdependence is briefly discussed.

## 3.1.1 Stress

Imagine an object oriented in the cartesian coordinate system with a number of forces acting on it, such that the vector sum of all the forces is zero. Take a slice orthogonal to the -direction and define a small area on this slice as . Let the total force acting on this small area be

 (3.1)

We can define the following scalar quantities.

 (3.2)

The subscripts and in refer to the plane and the force direction, respectively. Similarly considering slices orthogonal to the and -directions, we obtain

 (3.3)

and

 (3.4)

The scalar quantities can be arranged in a matrix form to yield the stress tensor

 (3.5)

The condition of static equilibrium implies .

Figure 3.1: Components of the stress tensor.

## 3.1.2 Strain

A body under elastic deformation experiences an internal restoring force. The amount of deformation caused is called strain and the corresponding force, the stress. Consider a pair of points at locations and which are deformed to locations and . The absolute squared distance between the deformed points can be written as

 (3.6)

Assuming to be a small displacement, a Taylor expansion about the point gives the absolute squared distance as

 (3.7)

Since the first term in (3.7) denotes the original squared distance between the points, the change in the squared distance becomes

 (3.8) (3.9)

 (3.10)

Here, denote the components of the strain tensor and are defined as

 (3.11)

Assuming that the second order term in (3.11) can be neglected and the resulting tensor is

 (3.12)

Note that in literature, engineering shear strain components, , are commonly used rather than the shear strain components described by (3.12). The relation is,

 (3.13)

Arranging the strain components (3.12) in matrix form gives the strain tensor

 (3.14)

The sign convention adopted for stress is that tensile stress causes an expansion, whereas compressive stress causes a contraction.

## 3.1.3 Stress-Strain Dependence

The relation between stress and strain was first identified by Robert Hook [Hook78]. Hook's law of elasticity is an approximation which states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). The most general relationship between stress and strain can be mathematically written as

 (3.15)

Here, is a fourth order elastic stiffness tensor comprising 81 coefficients. However, depending on the symmetry of the crystal the number of coefficients can be reduced. For cubic crystals such as Si and Ge only three unique coefficients , and , exist. These coefficients are known as the stiffness constants. To simplify the notations, the stress and strain tensor can be written as vectors using the contracted notations

 (3.16)

and the generalized Hook law in matrix form as

 (3.17)

 Si Ge

Table 3.1: Elastic stiffness constants in GPa [Levinstein99] and elastic compliance constants in m/N.

Of practical interest is the strain arising from a certain stress condition. The strain components can be obtained by inverting Hook's law and utilizing the compliance coefficients, .

 (3.18)

The stiffness and compliance tensors are linked through the relation . Using this relation, the three independent compliance coefficients can be calculated as

The compliance coefficients for Si and Ge, together with the stiffness coefficients are listed in Table 3.1. It is interesting to note that traditionally the stiffness coefficients are denoted by , while the compliance coefficients are denoted by .

## 3.1.4 Miller Indices

Figure 3.2: Planes in the cubic system with the Miller indices marked.

The Miller indices, denoted as , and , are a symbolic vector representation for the orientation of atomic planes and directions in a crystal lattice. Defining three lattice vectors forming the lattice axes, any crystal plane would intersect the axes at three distinct points. The Miller indices are obtained by taking the reciprocal of the intercepted values. By convention, negative indices are written with a bar over the indices. In Fig. 3.2, three planes in the cubic system, along with their Miller indices are shown, where the following nomenclature is adopted [Davies98].
• represents a direction
• denotes equivalent directions
• represents a plane with the normal vector
• denotes equivalent planes

## 3.1.5 Coordinate Transformation

Figure 3.3: Stress direction relative to the crystallographic coordinate system .

It is often required to know the stress in the crystallographic coordinate system for a stress applied along a general direction. Consider a generalized direction in which the stress is applied. The stress in the crystallographic coordinate system can be calculated using the transformation matrix .

 (3.19)

Here denotes the polar and the azimuthal angle of the stress direction relative to the crystallographic coordinate system, as shown in Fig. 3.3. The stress in the crystallographic coordinate system is then given by

 (3.20)

Applying a non-zero stress of magnitude applied along the [100], [110] and [111] directions, the stress tensors in the principal coordinate system read, respectively

 (3.21)

From (3.18), the corresponding strain tensors can be determined.

 (3.22)

 (3.23)

 (3.24)

## 3.1.6 Strain from Epitaxy

For the case of epitaxially grown Si on SiGe, the in-plane strain is given by (2.1). Consider an interface (primed) coordinate system, the -axis of which is perpendicular to the Si/SiGe interface. The strain tensor in this coordinate system can be written as

 (3.25)

with . Since epitaxial growth does not produce any in-plane shear strain in the interface coordinate system, we have

 (3.26)

The other three independent strain components, ( ) can be determined as described below [Hinckley90].

Since the strain is applied uniformly to the Si layer, all external stress components in the vertical direction vanish, . Therefore, using Hook's law stated in (3.15) we have

 (3.27)

where summation over repeated indices is implied. Expanding (3.27) gives

 (3.28)

which can be expressed in matrix form as

 (3.29)

The matrix elements can be determined from the elastic stiffness tensor through the relation,

 (3.30)

where denotes the transformation matrix in (3.19). Once the matrix elements are known, (3.29) can be inverted to determine the ( ). Having determined the strain tensor in the interface coordinate system, the tensor can be transformed to the principal coordinate systemusing

 (3.31)

 Substrate Orientation 0 0 0 0 0

Table 3.2: Components of the strain tensor for different orientations of the SiGe substrate.

Equations (3.29) and (3.31) can be solved to obtain the strain tensor for epitaxial growth of Si on an arbitrarily oriented SiGe substrate. Table 3.2 lists the expressions for the strain tensor components for the high-symmetry (001), (111) and (110) oriented SiGe substrates.

Next: 3.2 Structure of Relaxed Up: 3. Strain Effects on Previous: 3. Strain Effects on

S. Dhar: Analytical Mobility Modeling for Strained Silicon-Based Devices


text:
# Math Help - Statistics

1. ## Statistics

I started studying for an MBA degree and I was told it would be helpful to study Statistics.So I did. And now I am stuck

Summation(i=1,n) (Oi - Ei) squared divided by Ei

I need to use this formula to work out the frequencies expected under the hypothesis that the number of absentees is independant of the day of the week Below is the number of employees absent for a single day during a particular period of time
MOn(121absent), Tues (87absent) Weds (87 absent) Thur (91 absent) Fri (114 absent) TOTAL 500
Then I have to "test at the 5% level whether the differences in the observed and expected data are significant.

I want to be really dense and say "Eh??"

I started studying for an MBA degree and I was told it would be helpful to study Statistics.So I did. And now I am stuck

Summation(i=1,n) (Oi - Ei) squared divided by Ei

I need to use this formula to work out the frequencies expected under the hypothesis that the number of absentees is independant of the day of the week
No that's not what this formula is used for. This what we call a test statistic,
it's something we know (approximately) the distribution of. It is a function of
the data, and the data expected under the test hypothesis. The $O_i$'s are the
observed frequencies in the analysis bins (in this case days), and the $E_i$ are the
frequencies we would have expected if our test hypos thesis were true.

Below is the number of employees absent for a single day during a particular period of time
MOn(121absent), Tues (87absent) Weds (87 absent) Thur (91 absent) Fri (114 absent) TOTAL 500
Then I have to "test at the 5% level whether the differences in the observed and expected data are significant.
Our test hypothesis is that there is no difference between the absentee rate
for differing days of the week. Under this hypothesis we expect the same
number of absentees on each day, with a total number of $500$ absentees.

So we have expected numbers $E_i$ of $100, 100, 100, 100,100$ for $i=1..5$,
where $i=1$ denotes Monday, $i=2$ denotes Tuesday etc.

The observed frequencies $O_i$ are $121,87,87,91,114$, so now we have
everything we need to plug into the formula for the test statistic.

$
\sum_{i=1}^5 \frac{(O_i-E_i)^2}{E_i}=\frac{(121-100)^2}{100}+\frac{(87-100)^2}{100}
$

$
+\frac{(87-100)^2}{100}+\frac{(91-100)^2}{100}+\frac{(114-100)^2}{100}
$
so:
$
\sum_{i=1}^5 \frac{(O_i-E_i)^2}{E_i}=4.41+1.69+1.69+0.81+1.96=10.56
$

Now if I recall correctly the test statistic has a $\chi^2$ (Chi squared)
distribution, and $4$ degrees of freedom (number of bins minus one)

Now we look up the value of $\chi^2$ with $4$ degrees of freedom that 95%
of all observations should fall below, this is $9.49$ Thus we would expect
our test statistic to exceed this no more than 5% of the time. In this case
our test statistic does exceed this value so "at 5% level the differences in
the observed and expected data are significant".

RonL

3. WOW!!!

p.s. Thank you so very much!!