In this problem we work out step-by-step the procedure for checking an equivalence relation. Denote by $\mathbb{Z}$ the set of all integers. Declare that two integers $x, y$ are related if $x - y$ is an integer multiple of $3$. In symbols:

$$x \sim y \Longleftrightarrow 3 \textrm{ divides } x - y .$$
We want to check if this is an equivalence relation. That means we need to check if $\sim$ is
(1) Reflexive
(2) Symmetric
(3) Transitive

We begin with (1). This means checking to make sure that for all integers $x$, we have $x \sim x$. Recall the definition of $\sim$ for this problem and we see that this is equivalent to saying that for all integers $x$, we have that $( x - x )$ is an integer multiple of $3$.

Is this true? If so, enter Y; if not, enter an integer for which this is false.

Next, we check (2). This means checking to make sure that for all integers $x, y$, we have $x \sim y \Leftrightarrow y \sim x$. Unwind the definition of $\sim$ as we have done for (1) and we see that

$x \sim y \Longleftrightarrow$ $= 3 m$ for some integer $m$

$y \sim x \Longleftrightarrow$ $= 3 m$ for some integer $m$

Based on that, is (2) true? If so, enter Y; if not, enter a pair of integers for which this is false.


Finally, we check (3). This means checking to make sure that for all integers $x, y, z$, if $x \sim y$ and $y \sim z$ then $x \sim z$.

Is this true? If so, enter Y; if not, give a triple of integers for which this fails.

Finally, based on this calculation, is $\sim$ an equivalence relation on the set of integers? Enter Y or N.