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equivrelationskey2

# equivrelationskey2 - McCombs Math 381 Equivalence Relations...

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Unformatted text preview: McCombs Math 381 Equivalence Relations and Partitions 1 Formal Definition: Given a set A , and a relation R on set A , we say that R is an equivalence relation provided R is reflexive, symmetric and transitive. Important Vocabulary: Suppose R is an equivalence relation on set A . 1. Let P = P 1 , P 2 , P 3 ,... P n be a collection of non-empty subsets of A such that A = P 1 ! P 2 ! P 3 ! ... ! P n and P i ! P j = " for i ! j . We say that P is partition of set A . 2. Let x ! " # \$ R = a % A : aRx { } . We call x ! " # \$ R the equivalence class of element x with respect to R . In other words, an equivalence class is a subset containing all elements in set A that are related to one another by R . Important Theorem: (i) Every equivalence relation on a set A induces an associated partition P of set A . Moreover, the components of the partition are generated by the equivalence classes of the elements of set A . (ii) Every partition P of set A induces an equivalence relation on set A . Examples: 1. Let A = 1,2,3 { } . (i) List all possible partitions of A . Partition1 P 1 = 1,2,3 { } Partition2 P 1 = 1,2 { } , P 2 = 3 { } Partition3 P 1 = 1,3 { } , P 2 = 2 { } Partition4 P 1 = 2,3 { } , P 2 = 1 { } Partition5 P 1 = 1 { } , P 2 = 2 { } , P 3 = 3 { } (ii) Find the number of possible equivalence relations on...
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equivrelationskey2 - McCombs Math 381 Equivalence Relations...

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