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GATE CSE 2024 Session 1 Q32: Possible |A| with bijections A → B and A × A → A ∪ B

Question

Let AA and BB be non-empty finite sets such that there exist one-to-one and onto functions (i) from AA to BB and (ii) from A×AA \times A to A∪BA \cup B. The number of possible values of ∣A∣|A| is __________

Try it: numerical answer: type the value

Numerical answer. No negative marking.
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Answer: 2 (official GATE 2024 final answer key)

Concept used: A bijection between finite sets means equal cardinalities.

First step: With ∣A∣=∣B∣=n|A| = |B| = n, you need n2=∣A∪B∣n^2 = |A \cup B|, and n≤∣A∪B∣≤2nn \le |A \cup B| \le 2n.

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Source: official GATE 2024 Session 1 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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