What does it mean even? And odd?

Question

I'm actually an international student and I'm not very confident with specific mathematical terms. I was doing some exercises when I came up to the words even and odd. What do they mean exactly?

Here's the context:

Which of the following relations are even?

I. something II. something III. something

Results: (A) only I (B) only I and II (C) ... (D) ... (E) ...

Which of the following relations are odd?

I. something II. something III. something

Results: (A) only I (B) only I and II (C) ... (D) ... (E) ...

Thank you very much!

Answer 1

A (binary) relation $R \subseteq X \times Y$ is said to be even, if whenever $(x,y) \in R$ so is $(-x,y) \in R$. A binary relation is called odd, if whenever $(x,y) \in R$ so is $(-x,-y) \in R$.

Wasn't sure either at first, but the term is generalized from the usage in the study of functions (for a source see here).