Equivalence Relation with dividing integers

Question

Define $x\sim y$ if $4$ divides $(x+3y)$ for $x$ and $y$ integers. Show that is an equivalence relation.

Equivalence relation means it satisfies reflexity, symmetry, and transitivity.

Can anyone help me with symmetry and transitivity? Thank you!

Answer 1

Given that $x \sim y \iff x+3y \equiv 0 \mod 4$ for all $x, y \in \mathbb{Z}$

1) Reflexive property: We need to prove $x \sim x$ for all $x \in \mathbb{Z}$

$x+3x = 4x \equiv 0 \mod 4$

Hence, $x \sim x$

2) Symmetry property: We need to prove that $x \sim y \implies y \sim x$ for all $x, y \in \mathbb{Z}$

$x \sim y \implies x+3y \equiv 0 \mod 4 \implies 3(x+3y) \equiv 0 \mod 4 \implies y+3x \equiv 0 \mod 4 = y \sim x$

3) Transitive Property : We need to prove $x \sim y \land y \sim z \implies x \sim z$ for all $x, y, z \in \mathbb{Z}$

$x \sim y \land y \sim z \implies x+3y \equiv 0 \mod 4 \land y+3z \equiv 0 \mod 4$

$\implies (x+3y)+(y+3z) \equiv 0 \mod 4 \implies x+3z \equiv 0 \mod 4 = x \sim z $

Thus given relation is an Equivalence relation.

Answer 2

Note first that $x \sim y$ iff $x = y \pmod{4}$. A simple tabel of 16 options (for the classes mod 4 of $x$ and $y$) will do for that, e.g. or note that $3y = -y$ modulo $4$. Being an equivalence relation is then just noting that equality modulo $4$ is an equivalence relation.