equivalence relation

an equivalence relation on a set $\text{[math]}$ is a set $\text{[math]}$ of ordered pairs of elements of $\text{[math]}$ such that

$\text{[math]}$ for all $\text{[math]}$ (reflexive property).
$\text{[math]}$ implies $\text{[math]}$ (symmetric property).
$\text{[math]}$ and $\text{[math]}$ imply $\text{[math]}$ (transitive property).

[cite:;taken from @abstract_gallian_2021 chapter 0 preliminaries; definition equivalence relation]

when $\text{[math]}$ is an equivalence relation on a set $\text{[math]}$ , it is customary to write $\text{[math]}$ instead of $\text{[math]}$ . also, since an equivalence relation is just a generalization of equality, a suggestive symbol such as $\text{[math]}$ , $\text{[math]}$ , or $\text{[math]}$ is usually used to denote the relation. using this notation, the three conditions for an equivalence relation become $\text{[math]}$ ; $\text{[math]}$ implies $\text{[math]}$ ; and $\text{[math]}$ and $\text{[math]}$ imply $\text{[math]}$ . if $\text{[math]}$ is an equivalence relation on a set $\text{[math]}$ and $\text{[math]}$ , then the set $\text{[math]}$ is called the equivalence class of $\text{[math]}$ containing $\text{[math]}$ .
[cite:;from @abstract_gallian_2021 chapter 0 preliminaries]

the equivalence classes of an equivalence relation on a set $\text{[math]}$ constitute a partition of $\text{[math]}$ . conversely, for any partition $\text{[math]}$ of $\text{[math]}$ , there is an equivalence relation on $\text{[math]}$ whose equivalence classes are the elements of $\text{[math]}$ .
[cite:;from @abstract_gallian_2021 theorem 0.7 equivalence classes partition]

$\text{[math]}$ here $\text{[math]}$ is reflexive, antisymmetric and transitive so it is an equivalence relation

let $\text{[math]}$
first we check for reflexivity:
let $\text{[math]}$ then we know $\text{[math]}$ and therefore $\text{[math]}$ and therefore $\text{[math]}$ is reflexive
second we check for symmetry:
let $\text{[math]}$ so that $\text{[math]}$ and therefore by definition of $\text{[math]}$ we know $\text{[math]}$ , but only because $\text{[math]}$ doesnt mean $\text{[math]}$ because x might be 0, but we know $\text{[math]}$ because $\text{[math]}$ therefore $\text{[math]}$ therefore by definition of $\text{[math]}$ we get $\text{[math]}$ therefore $\text{[math]}$ is symmetric
third we check for transitivity:
let $\text{[math]}$ such that $\text{[math]}$ therefore by definition of the relation $\text{[math]}$ we get $\text{[math]}$ and we know $\text{[math]}$ therefore by definition of $\text{[math]}$ we get $\text{[math]}$ therefore $\text{[math]}$ is transitive
therefore $\text{[math]}$ is an equivalence relation