Relations & Their Properties: Selected Exercises
2 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.
3 20 Must an asymmetric relation be antisymmetric? A relation is asymmetric a b ( aRb (b, a) R ).
4 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. and defn of ) 3. a b ( ( aRb bRa ) a = b ) (Implication’s premise is false.) 4.Therefore, asymmetry implies antisymmetry.
5 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.
6 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.
7 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
8 40 List the 16 different relations on { 0, 1 }.
9 40 List the 16 different relations on { 0, 1 }. –A relation on {0, 1} is a subset of {0,1} x {0,1}. –{0,1} x {0,1} = { (0,0), (0,1), (1,0), (1,1) }. –There are 2 |{0,1} x {0,1}| = 2 |{0,1}||{0,1}| = 2 2x2 = 2 4 = 16 such subsets. –They are:
10 1. 2.{ (0,0) } 3.{ (0,1) } 4.{ (1,0) } 5.{ (1,1) } 6.{ (0,0), (0,1) } 7.{ (0,0), (1,0) } 8.{ (0,0), (1,1) } 9.{ (0,1), (1,0) } 10.{ (0,1), (1,1) } 11.{ (1,0), (1,1) } 12.{ (0,0), (0,1), (1,0) } 13.{ (0,0), (0,1), (1,1) } 14.{ (0,0), (1,1), (1,0) } 15.{ (1,1), (0,1), (1,0) } 16.{ (0,0), (0,1), (1,0), (1,1) }
11 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.
12 50 Prove R is antisymmetric R R -1 { ( a, a ) | a A }. We prove this implication by contradiction: 1.Assume R is antisymmetric: a b ( ( aRb bRa ) a = b ). 2.Assume it is not the case that R R -1 { ( a, a ) | a A }. 3. a b (a, b) R R -1, where a b. (Step 2) 4.Let (a, b) R R -1, where a b. (Step 3) 5.aRb, where a b. (Step 4) 6.aR -1 b, where a b. (Step 4) 7.bRa, where a b. (Step 6 & defn of R -1 ) 8.R is not antisymmetric, contradicting step 1. (Steps 5 & 7) 9.Thus, R is antisymmetric R R -1 { ( a, a ) | a A }.
13 50 continued Prove R R -1 { ( a, a ) | a A } R is antisymmetric. 1.Assume R R -1 { ( a, a ) | a A }. 2.Assume R is not antisymmetric: ¬ a b ( ( aRb bRa ) a = b ) 3. a b ( aRb bRa a b ) (Step 2) 4.bR -1 a, where a b. (Step 3 & defn. of R -1 ) 5.( b, a ) R R -1 where a b, contradicting step 1. (Step 3 & 4) 6.Therefore, R is antisymmetric.