Suppose X and Y are sets and |X| 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 A. |X| = 1, |Y| = 97 B. |X| = 97, |Y| = 1 C. |X| = 97, |Y| = 97 D. None of the above

GATE 1996 · Discrete Mathematics · Functions · medium

Answer: |X| = 1 and |Y| = 97 (option A), since |Y|^|X| = 97 and 97 is prime.

1. Write the function-count equation: The number of functions from X to Y is |Y|^|X|. Setting this equal to the given total gives |Y|^|X| = 97.
2. Solve using primality of 97: Since 97 is prime, the only way to write it as a^b with positive integers is 97 = 97^1. Hence |Y| = 97 and |X| = 1. Checking distractors: B gives 1^97 = 1 and C gives 97^97, neither equals 97, so only A survives.