How to symbolize mathematics statements

Question

The sum of two even integers is also even.

I assume P is the sum of two integers, Q is an even number. So $P \rightarrow Q$. Am I correct?

Answer 1

This is basically right, except $P$ and $Q$ should both be sentences. For example, let $P$ = "$N$ is the sum of two even integers" and let $Q$ ="$N$ is even". Then we can symbolize your statement with $P \to Q$.

Even this is not quite right though because we don't want $N$ to name a particular number. Rather, we'd like $N$ to be a variable that ranges over all the natural numbers. For this, we need first order logic, which you've indicated is not available to you.

Answer 2

No, because in your case, $P$ and $Q$ are not statements.

Mathematically, one would write "the su of two even integers is also even" like this:

$$\forall n,m\in 2\mathbb Z,\quad m+n\in 2\mathbb Z.$$

Answer 3

I think you're confusing propositions with variables. I'll make it simple – mostly because I'm not sure what you're talking about.

Propositions are basically statements that are either true or false, like "$2$ is even", "$3$ is even".

You can make propositions depend on a variable subject, like $P(x) :=$ "$x$ is even": these are called predicates. Note that the truth of a predicate depends on the value of the variable(s) it depends on.

You can also make new propositions from predicates by using quantifiers: one example could be $\exists x : P(x)$ (read: "some $x$'s verify $P(x)$ – true, in our case), another could be $\forall x : P(x)$ (read: "all $x$'s verify $P(x)$ – false, in our case).

A precise statement of what you want to convey could then be: $$\forall x,y\ \mathrm{numbers} : P(x) \land P(y) \Rightarrow P(x+y)$$

To be even more specific, you could talk about what kind of "numbers" you're actually talking about (in your case, the integers). Your proposition could then look like: $\forall x,y \in \mathbb Z \dots$