Question

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

A = { x ∈ Z: x is an integer multiple of 3 }
B = { x ∈ Z: x is a perfect square }
C = { 4, 5, 9, 10 }
D = { 2, 4, 11, 14 }
E = { 3, 6, 9 }
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|

Answer 1

(a) false

(b) true

(c) true

(d) true

(e) false

(f) true

(g) false

(h) true

(i) true

Explanation:

(a) 15 ⊂ A, since 15 is not a set, but an element, we cannot say of an element to be subset of a set. False

(b) {15} ⊂ A The subset {15} is a subset of A, since every element of {15}, that is 15, belongs to A.

15 ∈ {15} and 15 ∈ { x ∈ Z: x is an integer multiple of 3 } 15 is an integer multiple of 3. since 15/3=5. True

(c)∅ ⊂ A

∅ is a subset of any set. True

(d) A ⊆ A

A is a subset of itself. True

(e)∅ ∈ B

∅ is not an element, it is a subset, so it does not belong to any set. False

(f)A is an infinite set.

Yes, there are infinite integers multiple of 3. True

(g)B is a finite set.

No, there are infinite integers that are perfect squares. False

(h)|E| = 3

The number of elements that belong to E are 3. True

(i)|E| = |F|

The number of elements that belong to F are 3. So is the number of elements of E. True