Quotient Set

Topics: Discrete Mathematics - Relation

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(definition)

Let $R$ be an equivalence relation in $A$. We define the quotient set of $A$ on $R$ as:

$$A/R=\{[a]\mid a\in A\}$$

Example

Let $A=\{1,2,3,4\}$ and…

$$R=\{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(3,4),(4,3)\}$$

From this set and relation, we have the following equivalence classes:

• $[1]=\{1,2\}=[2]$
• $[2]=\{3,4\}=[4]$

So finally, we have that our quotient set of $A$ on $R$ is:

$$A/R=\{[1],[3]\}=\{[2],[4]\}$$