Let 𝐴 and 𝐵 be non-empty finite sets such that there exist one-to-one and…
2024
Let 𝐴 and 𝐵 be non-empty finite sets such that there exist one-to-one and onto functions (i) from 𝐴 to 𝐵 and (ii) from 𝐴 × 𝐴 to 𝐴 ∪ 𝐵. The number of possible values of |𝐴| is __________
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Correct answer: 2
Key idea: let n = |A| = |B| since there is a one-to-one onto map from A to B.
Then |A×A| = n^2. Also |A∪B| = |A| + |B| - |A∩B| = 2n - k, where k = |A∩B| and 0 ≤ k ≤ n.
A bijection between A×A and A∪B gives n^2 = 2n - k, so k = 2n - n^2.
Since k must satisfy 0 ≤ k ≤ n, test positive integers n: n = 1 gives k = 1 (valid); n = 2 gives k = 0 (valid); n ≥ 3 gives k ≤ -3 (impossible).
Thus n can be 1 or 2, so there are 2 possible values of |A|.
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