Questions: Subsets and Supersets

Question 1

Which of the following is true?

• A. {1, 2} ⊆ {1, 3, 5}
• B. {2, 4} ⊆ {1, 2, 3, 4, 5}
• C. 5 ⊆ {1, 2, 3, 4, 5}
• D. {1, 2, 3} ⊆ {1, 2}

Answer: (B)

{2, 4} ⊆ {1, 2, 3, 4, 5} because both 2 and 4 appear in the larger set. Option A fails because 2 is in {1, 2} but not in {1, 3, 5}. Option C is a type error: 5 is an element, not a set, so the ⊆ symbol does not apply (you would write 5 ∈ {1,2,3,4,5}). Option D fails because 3 is in {1, 2, 3} but not in {1, 2}.

Question 2

The empty set is a subset of every set.

• T. True
• F. False

Answer: True

To say ∅ ⊆ A, you must show that every element of ∅ is in A. Since ∅ has no elements, there are no elements to check — the condition is satisfied vacuously. This is the same logic as vacuous truth in conditional statements: 'If x ∈ ∅, then x ∈ A' is true because the hypothesis is never satisfied.

Question 3

Is {1, 2, 3} a proper subset of {1, 2, 3}? Explain why or why not.

Model answer: No. {1, 2, 3} is a subset of {1, 2, 3} (every element of the first is in the second), but not a proper subset because the sets are equal. A proper subset requires A ⊆ B and A ≠ B — meaning B must contain at least one element not in A.

Every set is a subset of itself (A ⊆ A always), but no set is a proper subset of itself (A ⊂ A is always false). The distinction between ⊆ (allows equality) and ⊂ (excludes equality) parallels the distinction between ≤ and < for numbers.