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Relations & Their Properties: Selected Exercises Exercise 10 Which relations in Exercise 4 are irreflexive? A relation is irreflexive a A (a, a) R. Ex. 4 relations on the set of all people: a) a is taller than b. b) a and b were born on the same day. c) a has the same first name as b. d) a and b have a common grandparent. Copyright © Peter Cappello 2011 2 Exercise 20 Must an asymmetric relation be antisymmetric? A relation is asymmetric a b ( aRb (b, a) R ). Copyright © Peter Cappello 2011 3 Exercise 20 Must an asymmetric relation be antisymmetric? A relation is asymmetric a b ( aRb (b, a) R ). To Prove: (a b ( aRb (b, a) R ) ) (a b ( (aRb bRa ) a = b ) ) Proof: 1. Assume R is asymmetric. 2. a b ( ( a, b ) R ( b, a ) R ). (step 1. & defn of ) 3. a b ( ( aRb bRa ) a = b ) (implication premise is false.) 4. Therefore, asymmetry implies antisymmetry. Copyright © Peter Cappello 2011 4 Exercise 20 continued Must an antisymmetric relation be asymmetric? (a b ( ( aRb bRa ) a = b ) ) a b ( aRb ( b, a ) R )? Work on this question in pairs. Copyright © Peter Cappello 2011 5 Exercise 20 continued Must an antisymmetric relation be asymmetric ? (a b ( (aRb bRa ) a = b ) ) a b ( aRb (b, a) R ) ? Proof that the implication is false: 1. Let R = { (a, a) }. 2. R is antisymmetric. 3. R is not asymmetric: aRa (a, a) R is false. Antisymmetry thus does not imply asymmetry. Copyright © Peter Cappello 2011 6 Exercise 30 • Let R = { (1, 2), (1, 3), (2, 3), (2, 4), (3, 1) }. • Let S = { (2, 1), (3, 1), (3, 2), (4, 2) }. • What is S R? R SR S 1 1 2 2 3 4 3 4 Copyright © Peter Cappello 2011 7 Exercise 50 Let R be a relation on set A. Show: R is antisymmetric R R-1 { ( a, a ) | a A }. To prove: 1. R is antisymmetric R R-1 { ( a, a ) | a A } We prove this by contradiction. 2. R R-1 { ( a, a ) | a A } R is antisymmetric. We prove this by contradiction. Copyright © Peter Cappello 2011 8 Exercise 50 Prove R is antisymmetric R R-1 { ( a, a ) | a A }. 1. Proceeding by contradiction, we assume that: 1. R is antisymmetric: a b ( ( aRb bRa ) a = b ). 2. It is not the case that R R-1 { ( a, a ) | a A }. 2. a b (a, b) R R-1, where a b. 3. Let (a, b) R R-1, where a b. (Step 2) 4. aRb , where a b. (Step 3) 5. aR-1b, where a b. (Step 3) 6. bRa, where a b. 7. R is not antisymmetric, contradicting step 1. (Steps 4 & 6) 8. Thus, R is antisymmetric R R-1 { ( a, a ) | a A }. Copyright © Peter Cappello 2011 (Step 1.2) (Step 5 & defn of R-1) 9 Exercise 50 continued Prove R R-1 { ( a, a ) | a A } R is antisymmetric. 1. Proceeding by contradiction, we assume that: 1. R R-1 { ( a, a ) | a A }. 2. R is not antisymmetric: ¬a b ( ( aRb bRa ) a = b ) 2. Assume a b ( aRb bRa a b ) 3. bR-1a, where a b. (Step 2s & defn. of R-1) 4. ( b, a ) R R-1 where a b, contradicting step 1. (Step 2 & 3) 5. Therefore, R is antisymmetric. Copyright © Peter Cappello 2011 (Step 1.2) 10