Questions: Set Operations and Venn Diagrams (Formal)

Question 1

In a two-set Venn diagram with sets A and B inside universal set U, how many distinct regions exist?

• A. 2
• B. 3
• C. 4
• D. 5

Answer: (C)

Four regions: (1) elements in A only (in A but not B), (2) elements in B only (in B but not A), (3) elements in both A and B (the overlap), and (4) elements in neither A nor B (outside both circles but inside the rectangle). Every element of U falls into exactly one of these four regions.

Question 2

De Morgan's Law states that (A ∪ B)ᶜ = Aᶜ ∪ Bᶜ.

• T. True
• F. False

Answer: False

De Morgan's Law states (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ — the complement of a union is the INTERSECTION of the complements, not the union. In a Venn diagram, (A ∪ B)ᶜ is the region outside both circles, which is exactly where you are not in A AND not in B — that is Aᶜ ∩ Bᶜ.

Question 3

Using a Venn diagram or set reasoning, explain why A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Model answer: An element x is in A ∩ (B ∪ C) when x ∈ A and x ∈ B ∪ C. The second condition means x ∈ B or x ∈ C. Case 1: x ∈ A and x ∈ B, so x ∈ A ∩ B. Case 2: x ∈ A and x ∈ C, so x ∈ A ∩ C. Either way, x ∈ (A ∩ B) ∪ (A ∩ C). The reverse inclusion works similarly. This is the distributive law: intersection distributes over union.

This identity is the set-theoretic version of the distributive law in algebra (a × (b + c) = a×b + a×c) and in logic (P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)). The parallel between sets, algebra, and logic reinforces that these are the same structural relationships appearing in different mathematical contexts.