24641
I'm reading a mathematics textbook, and there are a number of sentences which
end with もの or こと.

I can work out the intended meaning no problem, so what I would like explained
to me, are the rules for when I can use this grammatical construction.

This seems to happen in definitions, especially if mathematically written
conditions are involved.

For example:

> すなわち
>
>   1. m1, m2 ∈ N ⇒ m1 + m2 ∈ N, さらに 0 ∈ N
>
>   2. r ∈ R, n ∈ N ⇒ r•n ∈ N
>
>

>
> となるもの。

or

> f: S → T 全射 (surjection, epimorphism) であるとは、f (S) = T が成立すること。

I asked a (non-Japanese) mathematician, who told me that this can be used to
give commands, e.g., 勉強すること。However, I don't see why a command would appear in
definitions in this way...
CC BY-SA 3.0
2015-05-28T11:33:57.120
24606
2015-06-30T17:58:33.610
2020-06-17T08:18:27.500
-1
10202
question
nominalization mathematics
Sentence ending with こと or もの in mathematics definitions
I looked up "成立" and two of the meanings are "coming into existence;
conclusion" while having the meaning "to hold true" as a suru verb. I think
"成立すること" is simply to address the multiple meanings of the word, and it also
refers to the noun (f(S)=T) possessing the quality of the verb (成立する) by
nominalizing the verb (こと).

I haven't read a math book in a while, but if I can still remember
mathematicians parlance (haha) I would translate it as: "When f:S→T is a
surjection, f(S)=T still holds true."
CC BY-SA 3.0
2015-05-30T01:38:21.243
24635
2015-06-30T17:01:50.003
2015-06-30T17:01:50.003
10215
24606
answer
The sentence-final copula `である` ("be") is almost always omitted because it's
obvious in definitions, leaving the sentences looking like ending with nouns.
Both `もの` and `こと` are frequently used nominalizers translating "what do ~"
and "doing ~" respectively.

> すなわち、1. (...) 2. (...) となるもの。  
>  _i.e. **what** satisfies 1. (...) and 2. (...)._

* * *

> f:S→T が全射であるとは、f(S)=T が成立すること。  
>  _f:S→T is a surjection means **that** f(S)=T holds true._  
>  less literally, _f:S→T is a surjection when f(S)=T holds true._
CC BY-SA 3.0
2015-05-30T07:50:26.680
24641
2015-05-30T08:11:35.010
2015-05-30T08:11:35.010
7810
7810
24606
answer