Equivalence relation

Let $S$ be a set. A relation $\sim$ on $S$ is said to be an equivalence relation if $\sim$ satisfies the following three properties:

1. For every element $x \in S$ , $x \sim x$ . (Reflexive property)

2. If $x, y \in S$ such that $x \sim y$ , then we also have $y \sim x$ . (Symmetric property)

3. If $x, y, z \in S$ such that $x \sim y$ and $y \sim z$ , then we also have $x \sim z$ . (Transitive property)

Some common examples of equivalence relations:

The relation $=$ (equality), on the set of real numbers.
The relation $\cong$ (congruence), on the set of geometric figures in the plane.
The relation $\sim$ (similarity), on the set of geometric figures in the plane.
For a given positive integer $n$ , the relation $\equiv$ (mod $n$ ), on the set of integers. (Congruence mod $n$ )

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