Explanation Given information: Let A,B and 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: Consider the following statement, “For all sets A , B and C ( A − B ) ∩ ( C − B ) = A − ( B ∪ C ) ” The objective is to prove the above statement if it is true else provide a counter example. The above statement is false. Observe the following counter example, Let A = { x , y , z } , B = { y , z , w , q } , C = { x , z , p } be the subsets of a universal set U . Then, A − B = { x , y , z } − { y , z , w , q } = { x } C − B = { x , z , p } − { y , z , w , q } = { x , p } Thus, ( A − B ) ∩ ( C − B ) = { x } ∩ { x , p } = { x }

5th Edition

ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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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 7ES

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 U.

For all sets A,B and C.

(A−B)∩(C−B)=A−(B∪C)

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Chapter 6.3, Problem 6ES

Chapter 6.3, Problem 8ES

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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 U . For all sets A , B and C . ( A − B ) ∩ ( C − B ) = A − ( B ∪ C )

Solution Summary: The author explains that the objective is to prove the above statement if it is true or provide a counterexample.

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 U.

For all sets A,B and C.

(A−B)∩(C−B)=A−(B∪C)

Want to see the full answer?

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 U . For all sets A , B and C . ( A − B ) ∩ ( C − B ) = A − ( B ∪ C )

Solution Summary: The author explains that the objective is to prove the above statement if it is true or provide a counterexample.

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 U. For all sets A,B and C. (A−B)∩(C−B)=A−(B∪C)

Want to see the full answer?

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 U.

For all sets A,B and C.

(A−B)∩(C−B)=A−(B∪C)