Question

Suppose $X$ and $Y$ are sets and $|X|$ and $|Y|$ are their respective cardinalities. 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

Answer 1

The correct option is A $|X|=1,|Y|=97$

$X$ and $Y$ are sets. The cardinalities of $X$ and are $|X|$ and $|Y|$ respectively.
The number of functions from $X$ to $Y$ $=(|Y|{)}^{|X|}$
Given that number of functions from $X$ to $Y$ $=9$
$\therefore 97=(|Y|{)}^{|X|}$
So above implies that $|X|=1$ and $|Y|=97$