Use an element argument to prove each statement in 7-22. Assume that all sets are subsets of a universal set U. For all sets A , B , and C , ( A − B ) ∪ ( c − B ) = ( A ∪ C ) − B .

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To determine To prove: For all sets A ,B and C, (A-B) ∪ ( C − B ) = ( A ∪ C ) − B Explanation Given information: A and B be two sets. Concept used: ∪ : U n i o n of sets ∩ : I n t e r sec t i o n of sets Calculation: Let A , B and C are any three sets. Prove that ( A − B ) ∪ ( C − B ) = ( A ∪ C ) − B First prove that ( A − B ) ∪ ( C − B ) ⊆ ( A ∪ C ) − B . Suppose there is an element x is ( A − B ) ∪ ( C − B ) , by definition of union, x ∈ A − B or x ∈ C − B Case 1. If x ∈ A − B then by set difference x ∈ A and x ∉ B . Since x ∈ A by definition of union it follows that x ∈ A ∪ C . Thus, x ∈ A ∪ C and x ∉ B the definition of set difference implies that x ∈ ( A ∪ C ) − B . Hence, x ∈ A − B ⇒ x ∈ ( A ∪ C ) − B . Case 2. If x ∈ C − B then by set difference x ∈ C and x ∉ B . Since x ∈ C , by definition of union it follows that x ∈ A ∪ C . Thus, x ∈ A ∪ C and x ∉ B , the definition of set difference implies that x ∈ ( A ∪ C ) − B . Hence x ∈ C − B ⇒ x ∈ ( A ∪ C ) − B Therefore, both the cases yields x ∈ ( A ∪ C ) − B

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Answer 1

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Discrete Mathematics With Applicat...

5th Edition

EPP + 1 other

Publisher: Cengage Learning,

ISBN: 9781337694193

Solutions

Chapter

Section

Answer 3

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Chapter

Section

Answer 4

Chapter

Section

Answer 5

To determine

To prove:

For all sets A ,B and C, (A-B)∪(C−B)=(A∪C)−B

Explanation of Solution

Given information:

A and B be two sets.

Concept used:

∪:Union of sets∩:Intersection of sets

Calculation:

Let A,B and C are any three sets.

Prove that (A−B)∪(C−B)=(A∪C)−B

First prove that (A−B)∪(C−B)⊆(A∪C)−B.

Suppose there is an element x is (A−B)∪(C−B), by definition of union,

x∈A−B or x∈C−B

Case 1.

If x∈A−B then by set difference x∈A and x∉B.

Since x∈A by definition of union it follows that x∈A∪C.

Thus, x∈A∪C and x∉B the definition of set difference implies that x∈(A∪C)−B.

Hence, x∈A−B⇒x∈(A∪C)−B.

Case 2.

If x∈C−B then by set difference x∈C and x∉B.

Since x∈C, by definition of union it follows that x∈A∪C.

Thus, x∈A∪C and x∉B, the definition of set difference implies that x∈(A∪C)−B.

Hence x∈C−B⇒x∈(A∪C)−B

Therefore, both the cases yields x∈(A∪C)−B

Use an element argument to prove each statement in 7-22. Assume that all sets are subsets of a universal set U. For all sets A , B , and C , ( A − B ) ∪ ( c − B ) = ( A ∪ C ) − B .

Discrete Mathematics With Applicat...

Solutions

Discrete Mathematics With Applicat...