Equivalence (same cardinality) vs. equality — pick the non-equivalent pair: Which pair of sets is not equivalent (i.e., does not have the same number of elements)?

Introduction / Context:
Equivalent sets have the same cardinality (finite or infinite), regardless of the actual elements. We compare counts, not membership equality.

Given Data / Assumptions:

• Option (a): sizes 4 and 4
• Option (b): sizes 3 and 5
• Option (c): both empty → size 0 each
• Option (d): both countably infinite

Concept / Approach:
Check cardinalities directly; infinite even integers and infinite odd integers are in bijection, hence equivalent.

Step-by-Step Solution:
(a) Equivalent(b) Not equivalent (3 ≠ 5)(c) Equivalent(d) Equivalent (map n → n gives a bijection between even and odd forms)

Verification / Alternative check:
Construct a bijection for (d): 2n ↔ 2n + 1 is one-to-one and onto.

Why Other Options Are Wrong:
They have matching sizes or bijections; only (b) mismatches sizes.

Common Pitfalls:
Confusing “equivalent” with “equal”; they are different notions.

Final Answer:
A = {a, b, c}, B = {α, β, γ, δ, ν}