Sentence ending with こと or もの in mathematics definitions


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...




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._




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._

