Equivalence Classes Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy

List the equivalence classes of

\mathbb{Z}

under

a \sim b

iff

a \equiv b \pmod{2}

Solution

1
Class $[0]$ : all even integers $\{\ldots,-4,-2,0,2,4,\ldots\}$ .
2
Class $[1]$ : all odd integers $\{\ldots,-3,-1,1,3,5,\ldots\}$ .
3
These two classes partition $\mathbb{Z}$ .

Answer

[0]=\text{evens},\quad [1]=\text{odds}

Modulo 2 partitions the integers into exactly two classes: even and odd. This is the simplest non-trivial equivalence relation on

\mathbb{Z}

About Equivalence Classes

An equivalence class is the set of all elements that are related to a given element under an equivalence relation — it groups objects that are considered 'the same' in some specified sense.

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More Equivalence Classes Examples

Example 1 medium

Define the relation [formula] on [formula] by '[formula] iff [formula]'. Verify this is an equivalen

Example 2 medium

On the set of triangles, define [formula] if [formula] is congruent to [formula]. Show this is an eq

Example 4 medium

Define [formula] on functions from [formula] to [formula] by [formula] iff [formula]. Verify this is

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Set Equivalence Abstraction