Set Theory

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Question 1

If A and B are subsets of a universal set U such that A ∪ B = U and A ∩ B = ∅, which of the following statements is always true?

  • ∣A∣+∣B∣=∣U∣

  • ∣A∣=∣B∣

  • ∣A∣×∣B∣=∣U∣

  • None of the above

Question 2

Let A and B be two sets such that A ⊆ B. Which of the following statements is always true?

  • A′⊆B′

  • B′⊆A′

  • A′=B′

  • A′∪B′=U

Question 3

The number of non-trivial subsets of a set with 5 elements is

  • 32

  • 34

  • 30

  • 35

Question 4

If A = {x|x ∈ N and (x2 − 4)(x2 − 5) = 0} and B = {x|x ∈ l+ and x(x − 1)(x − 2) = 0} then (A ∪ B) - (A ∩ B) is

  • {1, 2}

  • {1}

  • {2}

  • 𝜙

Question 5

Which of the following is a subset of R (the set of real numbers)?

  • {x∣ x is a real solution to x2 + 1 = 0}

  • Q (the set of rational numbers)

  • {π, e}

  • All of the above

Question 6

Which of the following statements is false?

  • A−B=A∩B′

  • A∪(B∩C)=(A∪B)∩(A∪C)

  • A−(B∪C)=(A−B)∪(A−C)

  • A∩(B∪C)=(A∩B)∪(A∩C)

Question 7

If A and B are two sets such that A∪B = A×B, which of the following must be true?

  • Either A or B is empty

  • Both A and B are single-element sets

  • A and B are both empty

  • None of the above

Question 8

Which of the following is not a valid expression for the empty set ∅?

  • A − A', for any set

  • A ∩ A′, for any set A

  • A ∪ A′, for any set A

  • ∅ × A, for any set A

Question 9

Let A, B, and C be sets such that A∪B = A∪C and A∩B = A∩C. Which of the following must be true?

  • B = C

  • B ⊆ C

  • A ⊆ C

  • No conclusion can be drawn about B and C.

Question 10

The empty set ∅ is a subset of every set. Which of the following is false regarding the empty set?

  • ∅ ⊆ A for any set A.

  • ∅ ∈ A for any set A.

  • ∅ ∪ A = A

  • ∅ ∩ A = ∅

There are 10 questions to complete.

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