Equivalence Relations & Classes

Heavily interconnected with the idea of Relations:

What’s it all about? Eq Relations first:

If you want to say “Set $Q$ is an equivalence relation to Set $S$”, it needs to satisfy 3 properties for all $a$, $b$, $c$ in $S$.

1. Reflexivity: $a$ is related to itself, I.E. $a\equiv a$
2. Symmetry: If $a\equiv b$ then $b\equiv a$
3. Transitivity: If $a\equiv b$ and $b\equiv c$, then $a\equiv c$
   1. Note: There’s the idea of Transitive Closure of a Relation. If you have a set that only has (a, b) and (b, c), then the transitive closure is the set with all of those previous sets as well as indirect routes, like (a, c).

And Relations Classes?:

The idea of splitting different relations into different buckets (classes). Each bucket contains things considered the same by that rule.

Quick Example:

If your equivalence relation is $aRb$, where $R$ is $3|(m-n)$, then a suitable class would be where all remainders are 1. I.E. $[a]=\{x\in \mathbb{Z}|x=3k+1forsomek\in \mathbb{Z}\}$

Equivalent Relations and Classes

When the relation is all reflexive, symmetric and transitive. A class is splitting different relations into different buckets, containing things considered the same by that rule. E.G If your equivalence relation is $aRb$ where $R$ is $3|(m-n)$, then a suitable class would be where all remainders are 1. I.E. $[a]=\{x\in \mathbb{Z}|x=3k+1forsomek\in \mathbb{Z}\}$