In each of (a)-(f), answer the following question: Is A ⊆ B ? Is B ⊆ A ? Is either A or B a proper subset of the other? a. A = { 2 , { 2 } , ( 2 ) 2 } , B = { 2 , { { 2 } } } b. A = { 3 , 5 2 − 4 4 , 24 mod 7 } , B = { 8 mod 5 } c. A = { { 1 , 2 } , { 2 , 3 } } , B = { 1 , 2 , 3 } d. A = { a , b , c } , B = { { a } , { c } } e. A = { 16 , { 4 } } B = { 4 } f. A = { x ∈ R | cos x ∈ Z } , B = { x ∈ R| sin x ∈ Z }

Q: Expert Answer

To determine (a) Proof: Suppose A and B are any sets. To show that A − B ⊆ A , we must show that every element in __(1)__ is in __(2)__. But any element in A − B is in ___(3)___ and not in ___(4)___(by definition of A − B ). In particular, such an element is in A . Explanation Given information: A and B Are any sets. Concept used: Any element A − B is in A and not in B Calculation: Suppose A and B are two set. To show that A − B ⊆ A , need to show that if x ∈ A − B then x ∈ A . That is, every element in A − B is in A . From the definition of subtraction of sets x ∈ A − B ⇒ x ∈ A and x ∉ B . So any element in A − B is in A and not in B . By this, conclude that x ∈ A To determine (b) Suppose A and B are any sets and x ∈ A − B . [we must show that ( 1 ) ]. By definition of set difference, x ∈ ( 2 ) and x ∉ ( 3 ) . In particular, x ∈ ( 4 ) [which is what was to be shown].

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Answer 1

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Discrete Mathematics With Applicat...

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Publisher: Cengage Learning,

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Answer 2

BuyFind

Discrete Mathematics With Applicat...

5th Edition

EPP + 1 other

Publisher: Cengage Learning,

ISBN: 9781337694193

Solutions

Chapter

Section

Answer 3

Solutions

Chapter

Section

Answer 4

Chapter

Section

Answer 5

To determine

(a)

Proof: Suppose A and B are any sets. To show that A−B⊆A, we must show that every element in __(1)__ is in __(2)__. But any element in A−B is in ___(3)___ and not in ___(4)___(by definition of A−B ). In particular, such an element is in A.

Explanation of Solution

Given information:

A and B Are any sets.

Concept used:

Any element A−B is in A and not in B

Calculation:

Suppose A and B are two set.

To show that A−B⊆A, need to show that if x∈A−B then x∈A.

That is, every element in A−B is in A.

From the definition of subtraction of sets x∈A−B⇒x∈A and x∉B.

So any element in A−B is in A and not in B.

By this, conclude that x∈A

Answer 6

To determine

(b)

Suppose A and B are any sets and x∈A−B. [we must show that (1) ]. By definition of set difference, x∈(2) and x∉(3). In particular, x∈(4) [which is what was to be shown].

Discrete Mathematics With Applicat...

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Discrete Mathematics With Applicat...

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