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Equivalence relation

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A relation on set X is called an equivalence relation if it satisfies the following conditions:

  • Reflexivity: $A \sim A$ for all $A \in X$.
  • Symmetry: $A \sim B => B \sim A$ for all $A,B \in X$.
  • Transitivity: $A \sim B, B \sim C => A \sim C$ for all $A,B,C \in X$.

The equivalence class $[A]$ of $A\in X$ is the set of all elements equivalent to $A$: $$[A]=\{B\in X: B \sim A\}.$$

Theorem.

  • (1) Equivalence classes are disjoint.
  • (2) The union of all equivalence classes is the whole set X.

In other words this is a partition. See also quotient sets.

Examples:

In all of these examples, the equivalence relation respects the extra structure the set possesses (algebra, topology etc). To understand that see:

Two trivial examples, on any set:

  • $x \sim y$ for all $x,y$;
  • $x \sim y$ iff $x=y$;