Questions: Set Operations: Union, Intersection, and Complement

Question 1

Which of the following correctly states one of De Morgan's laws for sets?

• A. (A ∪ B)ᶜ = Aᶜ ∪ Bᶜ
• B. (A ∩ B)ᶜ = Aᶜ ∩ Bᶜ
• C. (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
• D. (A ∪ B)ᶜ = A ∩ B

Answer: (C)

De Morgan's first law: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ. Complementing a union flips it to an intersection. The logical reading makes this clear: 'not (P or Q)' means 'not P, and not Q.' The most common error is option A — students flip the outer operation correctly (complement becomes intersection) but forget to also complement A and B individually. Option D omits the complements entirely. The second De Morgan's law is (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ: complementing an intersection gives a union of complements.

Question 2

To prove that two sets A and B are equal (A = B), the standard method in proof-based mathematics is to:

• A. Show that A and B have the same number of elements
• B. Draw a Venn diagram showing the regions coincide
• C. Show A ⊆ B and B ⊆ A by taking an arbitrary element of each side and proving membership in the other
• D. Find a bijection between the elements of A and B

Answer: (C)

The standard method for proving set equality is mutual subset containment: take an arbitrary element x ∈ A, prove x ∈ B (establishing A ⊆ B), then take an arbitrary y ∈ B and prove y ∈ A (establishing B ⊆ A). This works by chasing through the logical definitions of the set operations. Venn diagrams (option B) build intuition but are not proofs. Counting elements (option A) only works for finite sets with known cardinalities. Bijections (option D) prove equal cardinality, not set equality — two different sets can have the same number of elements.

Question 3

De Morgan's laws for sets are independent rules that happen to resemble the logical De Morgan's laws — they require separate proofs.

• T. True
• F. False

Answer: False

De Morgan's laws for sets are not separate from the logical versions — they are the same rule applied to membership statements. The statement 'x ∈ (A ∪ B)ᶜ' means 'x ∉ A ∪ B', which means 'not (x ∈ A or x ∈ B)', which by the logical De Morgan's law means 'x ∉ A and x ∉ B', which means 'x ∈ Aᶜ and x ∈ Bᶜ', which means 'x ∈ Aᶜ ∩ Bᶜ'. The proof IS the logical law, applied through the definitions of the set operations. Set theory and logic are the same system expressed in two notations.

Question 4

For any two sets A and B, A ∪ B typically contains strictly more elements than A ∩ B.

• T. True
• F. False

Answer: False

This is false in the case where A = B. If A and B are identical sets, then A ∪ B = A ∩ B = A — they contain exactly the same elements. More generally, A ∪ B = A ∩ B if and only if A = B. The union contains 'at least as many' elements as the intersection (formally, A ∩ B ⊆ A ∪ B always holds), but 'strictly more' fails when the sets are equal. This is a good example of why 'usually true' intuitions need to be checked against edge cases in mathematical reasoning.

Question 5

How does the correspondence between set operations and logical connectives (∪ ↔ OR, ∩ ↔ AND, complement ↔ NOT) enable proofs about sets?

Model answer: The correspondence means that proving a membership statement about set operations reduces to applying logical rules to the definitions. To prove x ∈ A ∩ B, you prove 'x ∈ A and x ∈ B' — this IS the logical AND, so any logical rule for AND applies directly. De Morgan's laws, distributivity, and other set identities follow immediately from their logical counterparts because set membership statements just are logical propositions. This converts proof about sets into manipulating logical expressions, connecting set theory directly to the proof methods you already know.

The key insight is that set membership statements ('x ∈ A ∪ B') are literally logical claims ('x ∈ A or x ∈ B'), so the entire apparatus of propositional logic — equivalences, De Morgan's laws, distribution, contrapositive — transfers directly to set reasoning. This is why set operations are not just geometric intuitions drawn with Venn diagrams but a formal system where every identity has a proof derivable from logical axioms. Fluency with this correspondence is what enables rigorous set proofs rather than just diagram-based hand-waving.