How do I find the equivalence classes of these problems?

Question

This problem asks me to find the equivalence classes of the following relations ∼ on $\mathbb{Z}$:

• $m ∼ n \iff |m − 3| = |n − 3|$
• $m ∼ n \iff m + n$ is even

I just don't get how to do it. Thanks in advance for anyone who helps out.

Answer 1

To see how to do it, you need to understand the representation of equivalence classes. Given an equivalence relation ~ on S we denote the equivalence class of the representative x under ~ as $[x]=\{ y \in S | x\sim y \}$.

Let's start with the second equivalence relation: $m \sim n \Longleftrightarrow m + n $ even.

Equivalence classes form partitions of the given set.

We want to partition $\mathbb{Z}$ into disjoint sets, such that by applying the equivalence relation on elements of each set, we get back even numbers, and we want to find a representative for each. Intuitively, we will need to partition $\mathbb{Z}$ into the even and odd numbers, as the sum of an even and odd number is odd.

Expressing that in terms of our relation, we know that a number $x \in \mathbb{Z}$ is even if and only if $x \sim 0$, therefore all the even numbers can be represented by the equivalence class $[0]$. Similarly, x is odd if and only if $x \sim 1$ so the odd numbers are represented by $[1]$. These are the two equivalence classes for the given relation.

Now for the first one, we need to consider the numbers that are the same distance away from 3. These would be $ \{3\},\{2, 4\}, \{1, 5\}, \{0, 6\}$, {-1, 7}...