Partitions and equivalence relationship

partitions and equivalence relationship

An equivalence relation on a set X is a relation which is reflexive, symmetric and transitive Theorem: The set of all equivalence classes form a partition of X. Equivalence Relations. Definition Suppose that ∼ is a relation on a set X. Then i) ∼ is reflexive if ∀a ∈ X, a ∼ a, ii) ∼ is symmetric if ∀a, b ∈ X, a ∼ b . Equivalence relations and partitions. Indexed partitions. A family of sets (Ai)i∈I is called pairwise disjoint when any pair of them is disjoint: ∀i,j∈I, i≠j ⇒ Ai∩Aj.

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partitions and equivalence relationship

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