Equivalence Class

Topics: Discrete Mathematics - Algebra - Relation

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(definition)

Let $R$ be an equivalence relation in $A$. We define the equivalence class of $a\in A$:

$$[a]=\{x\in A\mid xRa\}$$

Example 1

Let $A=\{1,2,3,4,5\}$ and…

$$R=\{(1,1),(1,3),(1,5),(2,2),(2,4),(3,1),(3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(5,5)\}$$

…be an equivalence relation.

For each $a\in A$, we have that:

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

Notice that the equivalence classes of the elements in a given equivalence class are all the same.

Example 2

Let $A=\{a,b,c\}$ and $R=\{(a,a),(b,b),(c,c)\}$ be an equivalence relation.

We have that:

• $[a]=\{a\}$
• $[b]=\{b\}$
• $[c]=\{c\}$