Official PYQ · 1996Q1.3Sets, Relations and FunctionsMCQ1 mark

GATE CS 1996 Q1.3 — Sets, Relations and Functions

Suppose $X$ and $Y$ are sets and $|X| \text{ and } |Y|$ are their respective cardinality. It is given that there are exactly $97$ functions from $X$ to $Y$ . From this one can conclude that

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Answer: A

The total number of functions from a set $X$ to a set $Y$ is given by the formula:

\text{Number of functions} = |Y|^{|X|}

We are given that there are exactly $97$ functions from $X$ to $Y$ . Therefore, we have the equation:

|Y|^{|X|} = 97

Since $97$ is a prime number, it cannot be expressed as $m^n$ for any integers $m \ge 2$ and $n \ge 2$ . Thus, the only possible integer solution for the base and exponent is:

Exponent $|X| = 1$
Base $|Y| = 97$

This uniquely determines that $|X| = 1$ and $|Y| = 97$ .

Hence, the correct option is A.

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Published 10 Oct 2026

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