# Equivalences (me trying to understand an example)

1. Jun 10, 2014

### PsychonautQQ

1. The problem statement, all variables and given/known data
I'll be using ≈ as the equivalence symbol.

If m and n are integers, define m≈n to mean that m-n is even. Then ≈ is an equivalence on Z.

and
[0] = {x→Z | x≈0} is the set of even integers
[1] = {x→Z | x≈1} is the set of odd integers

where the x→Z was used to represent X is an element of Z.

So basically i'm having trouble making sense of this...
x≈0 means x-0, and if that has an equivalenet class of [0] somehow that means it's even? Can somebody try to explain what's going on here to me?

2. Jun 10, 2014

### CAF123

[0] and [1] are notations for equivalence classes. All elements of, say, [0] are related to 0 via the equivalence relation. This means that [0], for example, contains all integers, x, such that x ≈ 0. What does x≈0 mean? It means that for any x in the integers, x - 0 = x is even. Well this is simply the set of all even integers.

Does [1] now make sense?

3. Jun 10, 2014

### Fredrik

Staff Emeritus
Here, have an \in symbol: ∈

If you're going to use plain text, you can copy and paste that symbol instead of that rather confusing arrow. You can also use LaTeX. Type
Code (Text):
$[0]=\{x\in\mathbb Z|x\approx 0\}$
to get $[0]=\{x\in\mathbb Z|x\approx 0\}$. See the FAQ post for more information.

4. Jun 10, 2014

### HallsofIvy

Staff Emeritus
It means that x-0 is an even number and since x- 0= x, that is the same as saying x os an even number.