# Class Review: Class Equivalencies

So today in class we went through this whole concept of class equivalencies. It basically twists the whole idea of sets, elements, and how they’re related. The whole class I was just sitting there like – But I think I understand everything. Let’s see if I actually do or if I’m just bs’ing here.

An equivalence relation on a set S is a subset SxS satisfying the following R (relation):

• Reflexive property: (∀x∈S)((x,x)∈R). This means that every element relates to itself in some way.
• Symmetric property: (∀x,y∈S)((x,y)∈R ⇒ (y,x)∈R). This property guarantees that the relation between x and y, also relates y to x. Casey used the example of brothers. If I have a brother, I’m related to him and he’s related to me (thank god I don’t have a brother)
• Transitive property: (∀x,y,z∈S)([(x,y)+(y,z)∈R] ⇒ (x,z)∈R). The transitive property is like the symmetric property, but with three elements, sort of. It’s like saying my brother has a brother. I’m obviously related to both brothers (again thank god this is not true)

We can talk about this same relation in terms of wiggling.

wiggle relates pairs of elements in S, such that:

• Reflective: (∀x∈S)(x~x)
• Symmetric: (∀x,y∈S)(x~y ⇒ y~x)
• Transitive: (∀x,y,z∈S)(x~y + y~z ⇒ x~z)

There’s even more notation!

The equivalence class containing x.

Given an equivalence notation, ~, for some set, S:

[x] = {s∈S: x~s}. This is basically saying that in the set of [x], the elements signify the relation between x and elements in S.

Yeah, there’s EVEN MORE.

There’s this weird notation: S/~, which is defined as {[x]:x∈S}. This notation concerns sets of different equivalence classes.

This example in class helped me figure it out. We talked about groups earlier in the class, so we brought it back once again. Say the set S that we’re talking about is all the integers. The wiggle that relates x and y (x~y) is: x-y∈ nℤ. In this case, let’s say n=8.

ℤ/~ = {[x]: x∈ℤ} = {,,,,,,,}.

The set of  is all multiples of 8. When these number are divided by eight, there is no remainder, which is why you can designate it as 0. This is just a representative number for the set though and a very straightforward one at that. Any multiple of 8 can be chosen to represent this wiggle set, like -64, 8, 16, whatever.

The set of  includes all multiples of 8 that have a remainder 1. Numbers in this set include 17 (17/8 = 2 R1), 9 (9/8 = 1 R1), and 25 (25/8 = 3 R1), along with infinitely many more.

This is how all these sets are related to the integers.

We also went over another example that ended up giving us a donut, but we’ll go over that later. 