Equivalence relations and equivalence classes

Question

I dont know how to start this proof? Also, our professor did not explain equivalence classes fully so I am not understanding them very well.

Answer 1

You need to prove that $\sim$ is reflexive, symmetric, and transitive on $\mathbb Z$.

Then, an equivalence class of $a \in \mathbb Z$ is defined as $$[a] = \{b\mid b \in \mathbb Z \text { and } a\sim b\}$$: That is, $[a]$ is the class containing $a$ and all elements in $\mathbb Z$ that are related to $a$.

Hint: There are two equivalence classes that together, partition $\mathbb Z$.

Answer 2

In order to prove some relation $\sim$ is an equivalence relation on some set $A$ we need to show that for $a,b,c \in A$ we have the following properties:

• $a \sim a$
• $a \sim b \implies b \sim a$
• $a \sim b, b \sim c \implies a \sim c$

Now I will give hints to to each step for your equivalence relation:

• Does $2$ divide $0$?
• If $x$ is even, is $-x$ even?
• Since we know that $a-b, b-c$ are even, we know that $a-b = 2n, b-c=2k$ for some $k,n$

Now the equivalence class of some element $a$ are defined as:

$[a] = \{ b \mid a \sim b\}$

So think about the different elements of $\mathbb{Z}$, and what their equivlence classes will be. Hint: there are only 2 distinct classes.

Hope this helps