Questions: Complement of a Set

Question 1

If U = {a, b, c, d, e} and A = {a, c, e}, what is Aᶜ?

• A. {a, c, e}
• B. {b, d}
• C. {a, b, c, d, e}
• D. {}

Answer: (B)

Aᶜ contains all elements of U that are NOT in A. U has {a, b, c, d, e} and A has {a, c, e}, so the elements not in A are b and d. Therefore Aᶜ = {b, d}. Option A is A itself. Option C is U. Option D would mean every element of U is in A, which is not the case.

Question 2

For any set A with universal set U, A ∩ Aᶜ = ∅.

• T. True
• F. False

Answer: True

No element can be both in A and not in A simultaneously. Therefore the intersection of A and its complement is always empty. This is the set-theoretic version of the logical law of non-contradiction: a statement cannot be both true and false.

Question 3

Explain why the complement of a set depends on the universal set, using an example.

Model answer: Let A = {2, 4}. If U = {1, 2, 3, 4, 5}, then Aᶜ = {1, 3, 5}. But if U = {2, 4, 6, 8}, then Aᶜ = {6, 8}. The complement changes because it contains everything in U that is not in A, and different universal sets provide different 'everythings.'

The complement is not an absolute property of a set — it is relative to the universal set under consideration. This is why good mathematical writing always specifies the universal set before using complements. Without knowing U, 'the complement of A' is ambiguous.