I'm doing an advanced maths class for high school and we have just started a topic about proofs.
One of the questions (assume all numbers are integers here) is to prove that if $x\cdot y$ and $x + y$ are even, then both $x$ and $y$ are even.
I know that an even number is defined by 2m. With a proof like this, could I start my proof by assuming that x and y are even and then substituting into $x\cdot y$ and $x + y$? Or should I be starting from them and proving that $x$ and $y$ are even.
Alternatively, would it be better (or simpler) to prove starting by assuming that the numbers are all odd?
There are at least two simple ways to prove what you want to prove.
The most basicway to prove what you want to prove is to separate four possible cases:
If you can prove that cases $1$, $2$ and $3$ are impossible, then you have proven that $4$ is true.
Another way to prove is to first use the fact that $x+y$ is even. You can probably easily prove that if $x+y$ is even, then either $x$ and $y$ are both odd or they are both even. This is simple to prove since its equivalent statement, the statement "if one of $x,y$ is odd and the other is even, then $x+y$ is odd" is simple to prove.
Once you have shown that $x$ and $y$ are both odd or both even, you just have to show that they cannot both be odd, which is simple since if they are both odd, then $xy$ is odd.
You can also start from the other end and use the fact that $xy$ is even, since that means that at least one of the numbers is even. Then you prove that the other cannot be odd as that yould mean that $x+y$ is odd.