Exercise 3.1.2: Set membership and subsets - true or false, cont. Use the definitions for the sets given below to determine whether each statement is true or false: A = { x ∈ Z: x is an integer...

Exercise 3.1.2: Set membership and subsets - true or false, cont.

Use the definitions for the sets given below to determine whether each statement is true or false:

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  A = { x ∈ Z: x is an integer multiple of 3 }

  
  


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  B = { x ∈ Z: x is a perfect square }

  
  


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  C = { 4, 5, 9, 10 }

  
  


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  D = { 2, 4, 11, 14 }

  
  


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  E = { 3, 6, 9 }

  
  


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  F = { 4, 6, 16 }

  
  

An integer x is a perfect square if there is an integer y such that x = y2.

(a)15 ⊂ A

(b){15} ⊂ A

(c) ∅ ⊂ A

(d) A ⊆ A

(e) ∅ ∈ B

(f) A is an infinite set.

(g) B is a finite set.

(h) |E| = 3

(i) |E| = |F|

Exercise 3.1.3: Subset relationships between common numerical sets.

Indicate whether the statement is true or false.

(a) Z ⊂ R

(b) Z ⊆ R

(c) Z ⊆ R+

(d) N ⊂ R

(e) Z+ ⊂ N

Exercise 3.2.1: Sets of sets - true or false.

Let X = {1, {1}, {1, 2}, 2, {3}, 4 }. Which statements are true?

(a) 2 ∈ X

(b) {2} ⊆ X

(c) {2} ∈ X

(d) 3 ∈ X

(e) {1, 2} ∈ X

(f) {1, 2} ⊆ X

(g) {2, 4} ⊆ X

(h) {2, 4} ∈ X

(i) {2, 3} ⊆ X

(j) {2, 3} ∈ X

(k) |X| = 7

Exercise 3.3.1: Unions and intersections of sets.

Define the sets A, B, C, and D as follows:

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  A = {-3, 0, 1, 4, 17}

  
  


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  B = {-12, -5, 1, 4, 6}

  
  


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  C = {x ∈ Z: x is odd}

  
  


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  D = {x ∈ Z: x is positive}

  
  

For each of the following set expressions, if the corresponding set is finite, express the set using roster notation. Otherwise, indicate that the set is infinite.

(a) A ∪ B

(b) A ∩ B

(c) A ∩ C

(d) A ∪ (B ∩ C)

(e) A ∩ B ∩ C

(f) A ∪ C

(g) (A ∪ B) ∩ C

(h) A ∪ (C ∩ D)

Exercise 3.6.1: Cartesian product of three small sets.

The sets A, B, and C are defined as follows:

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  A = {tall, grande, venti}

  
  


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  B = {foam, no-foam}

  
  


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  C = {non-fat, whole}

  
  

Use the definitions for A, B, and C to answer the questions. Express the elements using n-tuple notation, not string notation.

(a) Write an element from the set A × B × C.

(b) Write an element from the set B × A × C.

(c) Write the set B × C using roster notation.