Determine the equivalence relation on {1, 2, 3, 4}

Question

If the relation is an equivalence relation, list the equivalence classes.

$$\{(x, y) : 4 \mid x - y\}$$

I have no clue how to solve this.

What I have tried is:

To know its an equivalence relation, it has to be reflexive, symmetric and transitive.

So it is reflexive as $(1,1), (2,2), (3,3), (4,4)$ is valid...

Any help is much appreciated...

To get curly braces to display, use \{ and \}. – Brian M. Scott Nov 26 '13 at 23:32 — Brian M. Scott, Nov 26 '13 at 23:32

score 0 · Answer 1 · answered Nov 26 '13 at 23:34

HINT:

Symmetry: If $4\mid x-y$, there is an integer $k$ such that $x-y=4k$. And $y-x=-(x-y)$, so ... ?
Transitivity: If $4\mid x-y$ and $\mid y-z$, then there are integers $k$ and $m$ such that $x-y=4k$ and $y-z=4m$. Try to use this to express $x-z$ as a multiple of $4$.

1 Answers1