2345_2019-Spring-HW5-Relations-Solutions.pdf - Math 2345 \u2013 Discrete Mathematics Homework 5 Relation Theory Solutions Due Date Wednesday(beginning of

2345_2019-Spring-HW5-Relations-Solutions.pdf - Math 2345...

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Homework 5: Relations Solutions Math 2345 – Discrete Mathematics Homework 5: Relation Theory Solutions Due Date: Wednesday, March 13, 2019 (beginning of class) Problem 5.1 (Relation Basics).(a) LetAbe the set of all strings of length three using only0’s,1’s, and2’s. Define a relationRonAas follows:R={(s,t) :the sum of characters insequals the sum of characters int}.i. Is(012,210)R?ii. Is(101,200)R?iii. Is(212,121)R?iv. List all images of102underR. (b) LetX={-1,0,1}andA=P(X), the power set ofX. LetRbe a relation defined onAas follows:R={(S, T) :the sum of all the elements inSequals the sum of all the elements inT}i. List three different elements ofR.ii. List an element fromA×Athat is not inR.iii. List all elements ofAthat areR-related to. Solution 5.1. (a) i. Yes ii. Yes iii. No iv. 102 , 120 , 210 , 201 , 012 , 021 , 111 (b) i. Answers will vary. ( { 0 } , { 1 , - 1 } ) , ( { 1 } , { 1 } ) , ( { 0 } , { 1 , 0 , - 1 } ) ii. Answers will vary. ( { 1 } , { 0 } ) iii. Skip, this was a poor question. Problem 5.2 (Visualizing Relations).Draw the relational digraph of each relation below:(a) LetR={(1,2),(2,1),(1,3),(3,1),(2,2)}(b) LetR={(1,1),(1,5),(2,2),(2,3),(2,6),(3,2),(3,3),(3,6),(4,4),(5,1),(5,5),(6,2),(6,3),(6,6)}.(c) LetR={(a, b) : 2|(a-b)}onA={5,6,7,8,9,10}.(d) LetR={(X, Y) :|X-Y|>0}onA=P({a, b}). Solution 5.2. (a)
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Homework 5: Relations Solutions 1 2 3 (b) 1 2 3 4 5 6 (c) 5 7 9 6 8 10 (d) { a, b } { a } { b }
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Homework 5: Relations SolutionsProblem 5.3 (What Properties Do I Have?).(a) LetRbe a relation onZdefined byR={(m, n) :m-nis odd}i. Determine ifRis reflexive, irreflexive, or neither. Explain.ii. Determine ifRis symmetric, antisymmetric, or neither. Explain.iii. Determine ifRis transitive or not. Explain.(b) For each relation in Problem 5.2, list the properties it has: (reflexive, symmetric, antisymmetric, transitive).If it fails to have a property, explain why.(c) The following “proof” claims that any binary relation on a nonempty set that is symmetric and transitive is
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