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)?
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AA = {2, 4, 6, 8}, B = {u, v, w, x}
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BA = {a, b, c}, B = {α, β, γ, δ, ν}
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CA = {}, B = ϕ
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DA = {x: x = 2n, n ∈ N}, B = {x: x = 2n + 1, n ∈ N}.
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ENone of these
Answer
Correct Answer: A = {a, b, c}, B = {α, β, γ, δ, ν}
Explanation
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}