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Equivalence&amp;operation

Equivalence&amp;operation - Equivalence Lecture 6...

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Equivalence Lecture 6 Discrete Mathematical Structures

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Equivalence Properties of Relation Reflexivity Symmetry Transitivity Equivalence Equivalence and Partition
Reflexivity Relation R on A is Reflexive if for all a A , ( a , a ) R Irreflexive if for all a A, ( a , a ) R Let A ={1,2,3}, R A × A { (1,1) ,(1,3), (2,2) ,(2,1), (3,3) } ? {(1,2),(2,3),(3,1)} ? {(1,2),(2,2),(2,3),(3,1)} ? R is reflexive relation on A if and only if I A R F IR Nothing

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Visualized Reflexivity a b c A ={ a,b,c } = 1 1 0 1 1 1 0 0 1 R M
Symmetry Relation R on A is Symmetric if whenever ( a , b ) R , then ( b , a ) R Antisymmetric if whenever ( a , b ) R and ( b , a ) R then a = b . Asymmetric if whenever ( a , b ) R then ( b , a ) R (Note: neither anti- nor a-symmetry is the negative of symmetry)

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Symmetry Let A ={1,2,3}, R A × A {(1,1),(1,2),(1,3),(2,1),(3,1),(3,3)} symmetric. {(1,2),(2,3),(2,2),(3,1)} antisymmetric. {(1,2),(2,3),(3,1)} antisymmetric and asymmetric. {(11),(2,2)} symmetric and antisymmetric. φ symmetric and antisymmetric, and asymmetric!
= 1 1 0 1 0 1 0 1 1 R M Visualized Symmetry A ={ a,b,c } a b c Every edge has its reverse edge m i,j = m j,i

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Visualized Antisymmetry A ={ a,b,c } a b c Except cycles, no edge has reverse edge. m i,j + m j,i ≤ 1 for i ≠ j = 1 0 0 1 0 1 0 0 1 R M
Visualized Asymmetry A ={ a,b,c } a b c No cycle, no edge has reverse edge. = 0 0 0 1 0 1 0 0 0 R M m i,i = 0 m i,j + m j,i ≤ 1 for i ≠ j

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Transitivity Relation R on A is Transitivity if whenever ( a , b ) R, ( b , c ) R, then ( a , c ) R Let A ={1,2,3}, R A × A {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,3)} is transitive {(1,2),(2,3),(3,1)} is not transitive. {(1,3)} ? φ ? Transitive!
Visualized Transitivity a b c A ={ a,b,c } = 1 0 0 1 0 0 1 1 1 R M R is transitive if and only if R n R for all n 1 m i,j =1 and m j,k = 1 then m i,k = 1

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What s Wrong? A wrong proof: if R is a symmetric and transitive relation on A , then R must be reflexive. Proof: For any a,b A , if ( a , b ) R , by the symmetry of R , ( b , a ) R ; since R is transitive, ( a , a ) R . So, R is reflexive.
Equivalence Relation Relation R on A is an equivalence relation if and only if it is reflexible, symmetric and transitive. Equality is a special case of equivalence relation. An example: R Z × Z , ( x , y ) R if and only if is integer, i.e. x y (mod 3) 3: modulus, a is congruent to b mod 3 3 y x -

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Equivalence Relation Let A = {1,2,3,4,5,6} R:{<1,1>,<2,2>, <6,6>,<1,4>,<2,5>, <3,6>, ,<6,3>} R(1), R(2), R(3), R(4), R(5), R(6)? R: + 3 How about + 3 in Z? R(100)?
Partition Generated by Equivalence Equivalence class : Let R is a equivalence relation on A , then given a A , R ( a ) is a equivalence class induced by R . Quotient set : Q ={ R ( x )| x A , and R is a equivalence on A } Quotient set is a partition: For any a A , a R (a) (remember that R is reflexible) For any a , b A ( a , b ) R if and only if R ( a )= R ( b ), and ( a , b ) R if and only if R ( a ) R ( b )= φ

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Equivalence induced by Partition A 1 A 6 A 5 A 4 A 3
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Equivalence&amp;operation - Equivalence Lecture 6...

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