GATE CSE 2024 | Set 1 | Question: 22

Question

Let $A$ and $B$ be non-empty finite sets such that there exist one-to-one and onto functions $\text{(i)}$ from $A$ to $B$ and $\text{(ii)}$ from $A \times A$ to $A \cup B$. The number of possible values of $\text{|A|}$ is ___________.

Answer 1

There exist one-to-one and onto function between X and Y implies $|X| = |Y|$.

Hence, $|A| = |B|$ and $|A|^2 = |A| + |B| - |A \cap B|$.

Can $|A| = 1$?

Yes, when $|A| = |B| = |A \cap B| = 1$.

Can $|A| = 2$?

Yes, when $|A| = |B| = 2, |A \cap B| = 0$.

For larger values of $|A|$, $|A|^2 > |A| + |B|$.

Therefore, only 2 values are possible for $|A|$.