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 __________
Answer
2
Explanation
A one-to-one and onto function (a bijection) exists between two finite sets exactly when they have the same number of elements. Let \(|A| = n \ge 1\).
- Condition (i) gives \(|B| = n\).
- Condition (ii) gives \(|A \cup B| = |A \times A| = n^{2}\).
The union of two \(n\)-element sets has between \(n\) elements (when \(A = B\)) and \(2n\) elements (when they are disjoint), so \(n \le n^{2} \le 2n\), which forces \(n \le 2\).
- \(n = 1\): take \(A = B = \{a\}\); then \(|A \cup B| = 1 = 1^{2}\).
- \(n = 2\): take disjoint \(A = \{a, b\}\) and \(B = \{c, d\}\); then \(|A \cup B| = 4 = 2^{2}\).
Both values occur, so \(|A|\) has 2 possible values.
Answer: 2