   # 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 } ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.1, Problem 1ES
Textbook Problem
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## 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 }

To determine

(a)

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

### Explanation of Solution

Given information:

A and B Are any sets.

Concept used:

Any element AB is in A and not in B

Calculation:

Suppose  A and B are two set.

To show that ABA, need to show that if xAB then xA.

That is, every element in AB is in A.

From the definition of subtraction of sets xABxA and xB.

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

By this, conclude that xA

To determine

(b)

Suppose  A and B are any sets and xAB. [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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