Equivalence Relations

a relation $\sim$ is an equivalence relation if it satisfies the following properties

1. Reflexivity $x\sim x$
2. Symmetry $x\sim y⟺y\sim x$
3. Transitivity $x\sim y\wedge y\sim z⟹x\sim z$

Equivalence relations cleanly partition sets. Partitions and equivalence relations are equivalent.

Congruence

Congruences exist to create Quotient Algebras which are sort of like building blocks to bigger algebras.

Congruences are Equivalence Relations which hold that for an algebra (applying to all operations)

if $a\sim b$ and $c\sim d$ then $a☿c\sim b☿d$

Partitions become their own algebras and steal an operation from the previous algebras. This is a quotient algebra.

E.X. $\mathbb{Z}$ is a ring The relation which makes two elements of $\mathbb{Z}$ equivalent if they are the same $mod2$ is a congruence. The quotient is $\mathbb{Z}mod2$. The partition has 2 blocks, the evens and the odds.