What does it mean to describe the partition defined by the equivalence class?

Question

Question: Is this relation an equivalence relation? If yes, describe the partition defined by the equivalence classes. Justify your answer: The domain is the set of all integers. xEy if x + y is even. (An integer z is even if z = 2k for some integer k.)

So for a relationship to be an equivalence relation it must be reflexive, symmetric, and transitive. How would I apply it to this problem? Also, I'm don't quite understand what it means by describing the partition.

Answer 1

If $x + y$ is even, is $y + x$ even?
Is $x + x$ even, for all $x$?
If $x + y$ is even, and $y + z$ is even, is $x + z$ even?

Proving the above will allow you to conclude that this is an equivalent relation. They are asking you to determine what the equivalence classes are. Every integer is in exactly one class, with all the other "related" integers. To what class does 7 belong? 4? -3?

You might also find this page helpful, especially the example of people with the same last name.

Answer 2

"Describing the partition" of a set A given equivalence relations R1...Rn means to define each of the equivalence classes in the partitioned set. Any members of A which do not belong to one of the equivalence classes form a final equivalence class: their equivalence relation is that they do not relate under any of the relations R1...Rn.

This example can be confusing because pairs of integers belong in an equivalence class (or not), rather than individual integers, so we must define A to be the set of all pairs of integers.

• Let A be the set of pairs of integers.
• xEy divides A into two equivalence classes:
  • Class 1: x and y have the same parity
  • Class 2: x and y do not have the same parity