Questions: Axiom of Extensionality

Question 1

According to the axiom of extensionality, which of the following pairs of sets are equal?

• A. {1, 2, 3} and {1, 2} — one is a subset of the other
• B. {1, 2, 3} and {3, 1, 2} — they contain exactly the same members
• C. {1, 2} and {1, 2, 2} — both contain 1 and 2
• D. Both B and C — extensionality makes order and multiplicity irrelevant

Answer: (D)

{3, 1, 2} and {1, 2, 3} have the same members, so by extensionality they are equal. {1, 2, 2} = {1, 2} because membership is binary — 2 either belongs or doesn't; listing it twice adds no information. So both B and C identify equal pairs. The axiom eliminates both ordering and multiplicity as features of set identity. Option A is wrong: {1, 2} ⊆ {1, 2, 3} but they are not equal because 3 ∈ {1, 2, 3} but 3 ∉ {1, 2}.

Question 2

You want to prove that two sets A and B are equal. Which proof strategy does the axiom of extensionality most directly license?

• A. Show that A and B were defined by the same rule or formula
• B. Show A ⊆ B and B ⊆ A (double inclusion)
• C. Show that |A| = |B| (they have the same cardinality)
• D. Construct an explicit bijection between A and B

Answer: (B)

Extensionality says A = B iff ∀x (x ∈ A ↔ x ∈ B). The double-inclusion method proves exactly this: A ⊆ B means every x ∈ A satisfies x ∈ B (the → direction), and B ⊆ A means every x ∈ B satisfies x ∈ A (the ← direction). Together they establish the biconditional. Option A (same definition) is not sufficient — two differently-defined sets can be equal. Options C and D prove equal cardinality, not equality; infinite sets with the same cardinality can be quite different sets.

Question 3

The axiom of extensionality guarantees that there is exactly one empty set.

• T. True
• F. False

Answer: True

Suppose ∅₁ and ∅₂ are both empty sets. Then vacuously, every member of ∅₁ is a member of ∅₂ (there are no members to check), so ∅₁ ⊆ ∅₂. Symmetrically, ∅₂ ⊆ ∅₁. By extensionality, ∅₁ = ∅₂. So any two empty sets are identical. This is a non-trivial consequence — without extensionality, nothing prevents having multiple distinct empty sets. Extensionality collapses all 'empty-looking' objects into one canonical empty set.

Question 4

The axiom of extensionality implies that multisets (bags) and sets are the same kind of mathematical object.

• T. True
• F. False

Answer: False

Multisets intentionally track element multiplicity: ⟨1, 1, 2⟩ and ⟨1, 2⟩ are distinct multisets. This violates extensionality as applied to sets, which declares that membership is binary — 1 either belongs or doesn't, and listing it twice changes nothing about a set's identity. Sets and multisets are different structures; the axiom of extensionality specifically defines sets as objects where multiplicity is irrelevant, distinguishing them from multisets, sequences, and other collection types.

Question 5

Why is extensionality considered a substantive axiom rather than a trivial definition? What would mathematics look like if two distinct 'empty sets' could exist?

Model answer: Extensionality is substantive because it makes a non-obvious claim: sets have no internal structure, hidden state, or identity beyond membership. Alternative collection types (multisets, sequences, labeled sets) are all mathematically coherent but violate extensionality. If two empty sets could exist, then the proof 'A ∩ B = ∅ and A ∩ C = ∅ implies B = C' would fail (both equal different empty sets). Double-inclusion proofs would be unsound. Uniqueness of constructions like intersections, power sets, and set-builder definitions would require separate axioms to guarantee.

The axiom is doing real work: it licenses the inference from 'same members' to 'same set,' which underlies virtually every proof of set equality in mathematics. Without it, set identity would need a separate criterion, and the entire framework of ZFC would need additional axioms to recover results we currently get for free. The existence of alternative frameworks (non-well-founded set theory, type theory with intensional equality) shows that extensionality is a genuine choice, not a logical necessity.