10B Week 10

# 10B Week 10 - CMPUT 272(Stewart Lecture 18 Reading Epp 8.5...

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CMPUT 272 (Stewart) Lecture 18 Reading: Epp 8.5 Partial Order Relations: reflexive, antisymmetric, and transitive Recall: A relation R on a set A is antisymmetric ⇔ ∀ x, y A , if ( x, y ) R and ( y, x ) R then x = y . Examples: 1. on R 2. on A , where A is a set of sets 3. | on Z + Theorem. The relation | on Z is not antisymmetric and therefore it is not a partial order relation. Proof: The integers 1 and -1 divide each other but are not equal. 2 1

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Theorem. The relation R on R × R where, for all ( a, b ) and ( c, d ) in R × R , ( a, b ) R ( c, d ) either a < c or both a = c and b d is a partial order relation. Proof: R is reflexive: Let ( a, b ) R × R . a = a and b b ; therefore ( a, b ) R ( a, b ). R is antisymmetric: Let ( a, b ) , ( c, d ) R × R such that ( a, b ) R ( c, d ) and ( c, d ) R ( a, b ). Then a c and c a , which implies that a = c . But now b d and d b , so b = d . Therefore ( a, b ) = ( c, d ). R is transitive: Let ( a, b ) , ( c, d ) , ( e, f ) R × R such that ( a, b ) R ( c, d ) and ( c, d ) R ( e, f ). Then a c and c e , which implies that a e .
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