Let A = { x ∈ Z | x = 6 a + 4 for some integer a } , B = { y ∈ Z | y = 18 b − 2 for some integer b } , and C = { z ∈ Z | z = 18 c + 16 for some integer c } . Prove or disprove each of the following statements. a. A ⊆ B b. B ⊆ A c. B = C

Q: Expert Answer

To determine (a) A ⊆ B Explanation Given information: Consider the sets. A = { x ∈ Z | x = 6 a + 4 for some integer a } B = { y ∈ Z | y = 18 b − 2 for some integer b } C = { z ∈ Z | z = 18 c + 16 for some integer c } Concept used: A ⊆ B means every element of A is in elements of B. Calculation: The objective is to prove or disprove the statement. A ⊆ B Assume that A ⊆ B is true, then every element in the set A is an element in the set B . Let x ∈ A , then there is an integer a such that x = 6 a + 4 and x is an element in B . Thus, there is an integer b such that x = 18 b − 2 . Since, x = 6 a + 4 and x = 18 b − 2 , then 6 a + 4 = 18 b − 2 Add 2 on both sides To determine (b) Prove or disprove B ⊆ A To determine (c) Prove or disprove B = C

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Answer 1

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Answer 2

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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

(a)

A⊆B

Explanation of Solution

Given information:

Consider the sets.

A={x∈Z|x=6a+4 for some integer a}B={y∈Z|y=18b−2 for some integer b}C={z∈Z|z=18c+16 for some integer c}

Concept used:

A⊆B means every element of A is in elements of B.

Calculation:

The objective is to prove or disprove the statement.

A⊆B

Assume that A⊆B is true, then every element in the set A is an element in the set B.

Let x∈A, then there is an integer a such that x=6a+4 and x is an element in B.

Thus, there is an integer b such that x=18b−2.

Since, x=6a+4 and x=18b−2, then

6a+4=18b−2

Add 2 on both sides

Answer 6

To determine

(b)

Prove or disprove B⊆A

Answer 7

To determine

(c)

Prove or disprove B=C

Let A = { x ∈ Z | x = 6 a + 4 for some integer a } , B = { y ∈ Z | y = 18 b − 2 for some integer b } , and C = { z ∈ Z | z = 18 c + 16 for some integer c } . Prove or disprove each of the following statements. a. A ⊆ B b. B ⊆ A c. B = C

Discrete Mathematics With Applicat...

Solutions

Discrete Mathematics With Applicat...

Let A = { x ∈ Z | x = 6 a + 4 for some integer a } , B = { y ∈ Z | y = 18 b − 2 for some integer b } , and C = { z ∈ Z | z = 18 c + 16 for some integer c } .

Prove or disprove each of the following statements.

a. A ⊆ B

b. B ⊆ A

c. B = C

Explanation of Solution

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