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# 17 - LECTURE 17 PROPERTIES OF EQUIVALENCE RELATIONS(8.4 Mi...

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LECTURE 17: PROPERTIES OF EQUIVALENCE RELATIONS ( § 8.4) Mi taku oyasin. (We are all related.) Lakota belief 1. Equivalence classes Let R be an equivalence relation on A . Let a A . The set [ a ] = { x A : xRa } is called the equivalence class of a . This is the set of all things related to a . Example 1. Let A = { 1 , 2 , 3 , 4 , 5 , 6 } , and R = { ((1 , 3) , (3 , 4) , (5 , 4) , (6 , 2) } . What do we need to add to R to make it an equivalence relation? Then what are the equivalence classes? We must add (1 , 1) , (2 , 2) , (3 , 3) , . . . , (6 , 6) to make the relation reflexive. We must add (3 , 1) , (4 , 3) , (4 , 5) , (2 , 6) to make the relation symmetric. We must then add (1 , 4) , (4 , 1) , (1 , 5) , (5 , 1) , (3 , 5) , (5 , 3) to make the relation transitive. Then the equivalence classes are [1] = { 1 , 3 , 4 , 5 } and [2] = { 2 , 6 } . Example 2. What are the equivalence classes under the operation = on Z ? (They are the one element sets.) Theorem 3. Let R be an equivalence relation on a nonempty set A . Let a, b A . We have [ a ] = [ b ] if and only if aRb . Proof. ( ) : Suppose [ a ] = [ b ]. Notice that since R is reflexive we have aRa . Thus a [ a ]. This means a [ b ]. Hence aRb .

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17 - LECTURE 17 PROPERTIES OF EQUIVALENCE RELATIONS(8.4 Mi...

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