GATE 2024 CS (CS1) – Question 32
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 $A\times A$ to $A\cup B$. The number of possible values of $|A|$ is __________
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Correct answer: 1
Explanation
(i) gives |A|=|B|=n. (ii) needs n² = |A∪B|, and since |A∪B| ≤ 2n we need n² ≤ 2n, so n ≤ 2. For n=1: A∪B has size 1 or... with n=2, |A∪B| must be 4 but max is 4 only if A,B disjoint; n=2 works (A∩B=∅) and n=1 works (A=B). Both give valid values; count of valid |A| is 2.