Explanation Given information: R is a relation defined on A = { 0 , 1 , 2 , 3 } . Let R = { ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 1 ) , ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 0 ) } . Calculation: A = { 0 , 1 , 2 , 3 } R = { ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 1 ) , ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 0 ) } The transitive closure of R needs to contain all ordered pairs in the relation R : { ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 1 ) , ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 0 ) } ⊆ R t The transitive closure of R is the smallest relation that contains R and that is transitive. Let us next determine which pairs to add using the transitive property. ( 0 , 1 ) ∈ R t and ( 1 , 1 ) ∈ R t ⇒ ( 0 , 1 ) ∈ R t ( 0 , 1 ) ∈ R t and ( 1 , 3 ) ∈ R t ⇒ ( 0 , 3 ) ∈ R t ( 0 , 2 ) ∈ R t and ( 2 , 2 ) ∈ R t ⇒ ( 0 , 2 ) ∈ R t ( 1 , 1 ) ∈ R t and ( 1 , 1 ) ∈ R t ⇒ ( 1 , 1 ) ∈ R t ( 1 , 1 ) ∈ R t and ( 1 , 3 ) ∈ R t ⇒ ( 1 , 3 ) ∈ R t ( 1 , 3 ) ∈ R t and ( 3 , 0 ) ∈ R t ⇒ ( 1 , 0 ) ∈ R t ( 3 , 0 ) ∈ R t and ( 0 , 1 ) ∈ R t ⇒ ( 3 , 1 ) ∈ R t ( 3 , 0 ) ∈ R t and ( 0 , 2 )

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ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

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Chapter 8.2, Problem 51ES

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The set Rt, transitive closure of R.

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Chapter 8.2, Problem 50ES

Chapter 8.2, Problem 52ES

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Chapter 8 Solutions

DISCRETE MATHEMATICS WITH APPLICATION (

Ch. 8.1 - If R is a relation from A to B, xA , and yB , the...Ch. 8.1 - Prob. 2TY Ch. 8.1 - Prob. 3TY Ch. 8.1 - Prob. 4TY Ch. 8.1 - If R is a relation on a set A, the directed graph...Ch. 8.1 - As in Example 8.1.2, the congruence modulo 2...Ch. 8.1 - Prove that for all integers m and n,m-n is even...Ch. 8.1 - The congruence modulo 3 relation, T, is defined...Ch. 8.1 - Define a relation P on Z as follows: For every...Ch. 8.1 - Prob. 5ES

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The set R t , transitive closure of R.

Solution Summary: The author explains that the transitive closure of R is the smallest relation that contains R.

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