Equivalence Class

https://youtu.be/TbCk79SoCYw

Equivalence class is the name given to a subset of some equivalence relation R which includes all elements that are equivalent to each other.

Let R be an equivalence relation on a set A. The set of all elements which are related to an element x of set A is called the equivalence class of x.

$[x]=\{y|(x,y)\in R\}$ is the equivalence class of x∈A

Example #

A={1,2,3,4,5} R = {(a,b) | a+b is even} R is an equivalence relation on set A.

• [1] = [3] = [5] = {1,3,5}
• [2] = [4] = {2,4}

Properties of Equivalence Class #

1. a∈[a]¹
   1. [a]={b|(a,b)∈R} and (a,a)∈R²
   2. ∴ aRa ⇒ and a∈[a]
2. b∈[a] then [a]=[b]³
   1. b∈[a]: bRa and aRb⁴
   2. If x∈[a] then xRa. xRa, aRb ⇒ xRb: x∈[b] ⇒ [a]⊆[b]
   3. If x∈[b] then xRb. xRb, bRa ⇒ xRa: x∈[a] ⇒ [b]⊆[a]
   4. By 2.,3. [a]=[b]
3. [a]∩[b]≠Φ then [a]=[b]⁵