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 ________

Let $|A| = n$, where $n$ is a positive integer ($n \ge 1$) since $A$ is non-empty and finite.

1. Condition (i): A bijection (one-to-one and onto function) exists from $A$ to $B$, which means:

   $$|B| = |A| = n$$

2. Condition (ii): A bijection exists from $A \times A$ to $A \cup B$, which implies:

   $$|A \times A| = |A \cup B|$$

   $$n^2 = |A \cup B|$$

3. By the Principle of Inclusion-Exclusion:

   $$|A \cup B| = |A| + |B| - |A \cap B| = 2n - |A \cap B|$$

   Since $0 \le |A \cap B| \le n$, we have the inequality:

   $$n \le |A \cup B| \le 2n$$

   Substituting $|A \cup B| = n^2$:

   $$n \le n^2 \le 2n$$

4. Dividing throughout by $n$ (since $n > 0$):

   $$1 \le n \le 2$$

Thus, the possible integer values for $n = |A|$ are $n = 1$ and $n = 2$:

• For $n = 1$: $|A \times A| = 1$, and if $A = B$, $|A \cup B| = 1$.

• For $n = 2$: $|A \times A| = 4$, and if $A \cap B = \emptyset$, $|A \cup B| = 4$.

Hence, there are 2 possible values of $|A|$.