Relations & Their Properties: Selected Exercises
Copyright © Peter Cappello2 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 Cappello3 Exercise 20 Must an asymmetric relation be antisymmetric? A relation is asymmetric a b ( aRb (b, a) R ).
Copyright © Peter Cappello4 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 Cappello5 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 Cappello6 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 Cappello7 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? RS S R
Copyright © Peter Cappello8 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 Cappello9 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. (Step 1.2) 3.Let (a, b) R R -1, where a b. (Step 2) 4.aRb, where a b. (Step 3) 5.aR -1 b, where a b. (Step 3) 6.bRa, where a b. (Step 5 & defn of R -1 ) 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 Cappello10 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 ) (Step 1.2) 3.bR -1 a, 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.