Explanation Given information: Let A,B a n d C be any set Concept used: ∪ : U n i o n of sets ∩ : I n t e r sec t i o n of sets ⊆ : subset of set Calculation: Suppose X and Y are two subsets of a universal set U . From the definition of differences and unions, The difference Y minus X is denoted by Y − X and is defined as the set of all elements that are present in Y but not in X . Also, the union X and Y is denoted by X ∪ Y and is defined as the set of elements that are present in at least either X or Y . Consider the statement, A ∪ ( B − C ) = ( A ∪ B ) − ( A ∪ C ) The above statement is false. Consider a counter example where A = { 1 , 2 , 3 } , B = { 3 , 4 , 5 } and C = { 2 , 4 , 6 } . Evaluate A ∪ ( B − C ) corresponding to the counter example. By the definition of difference. B − C = { 3 , 4 , 5 } − { 2 , 4 , 6 } = { 3 , 5 } Next find A ∪ ( B − C ) . By the definition of union. A ∪ ( B − C ) = { 1 , 2 , 3 } ∪ { 3 , 5 } = { 1 , 2 , 3 , 5 } Evaluate ( A ∪ B ) − ( A ∪ C ) corresponding to the counter example

5th Edition

ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

Chapter 6.3, Problem 13ES

To determine

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set ∪.

For all sets A,B and C ,

A∪(B−C)=(A∪B)−(A∪C)

Expert Solution & Answer

Chapter 6 Solutions

Discrete Mathematics With Applications

Ch. 6.1 - The notation is read”______” and means that___Ch. 6.1 - To use an element argument for proving that a set...Ch. 6.1 - Prob. 3TY Ch. 6.1 - An element x is in AB if , and only if,_______Ch. 6.1 - An element x in AB if, and only if,______Ch. 6.1 - An element x is in B-A if, and only if,______Ch. 6.1 - An elements x is in Acif, and only if.______Ch. 6.1 - The empty set is a set with ______Ch. 6.1 - The power set of a set A is _____Ch. 6.1 - Prob. 10TY

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To Prove: For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set ∪ . For all sets A , B and C , A ∪ ( B − C ) = ( A ∪ B ) − ( A ∪ C )

Solution Summary: The author explains how to prove each statement that is true and find a counterexample for each false statement. For all sets A,BandC, the difference Y minus

Concept explainers

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set ∪.

For all sets A,B and C ,

A∪(B−C)=(A∪B)−(A∪C)

Chapter 6 Solutions

To Prove: For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set ∪ . For all sets A , B and C , A ∪ ( B − C ) = ( A ∪ B ) − ( A ∪ C )

Solution Summary: The author explains how to prove each statement that is true and find a counterexample for each false statement. For all sets A,BandC, the difference Y minus

Concept explainers

To Prove: For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set ∪. For all sets A,B and C , A∪(B−C)=(A∪B)−(A∪C)

Chapter 6 Solutions

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set ∪.

For all sets A,B and C ,

A∪(B−C)=(A∪B)−(A∪C)