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Equivalence relation

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Let S be a set. A binary 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 \pmod n, on the set of integers. (Congruence mod n)
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