In general, there is no relationship between \(\card{B}\) and \(\card{f\inv(B)}\text{.}\) This is because \(B\) might contain elements that are not in the range of \(f\text{,}\) so we might even have \(f\inv(B) = \emptyset\text{.}\) On the other hand, there might be lots of elements from the domain that all get sent to a few elements in \(B\text{,}\) making \(f\inv(B)\) larger than \(B\text{.}\)

More specifically, if \(f\) is injective, then \(\card{B} \ge \card{f\inv(B)}\) (since every element in \(B\) must come from at most one element from the domain). If \(f\) is surjective, then \(\card{B} \le \card{f\inv(B)}\) (since every element in \(B\) must come from at least one element of the domain). Thus if \(f\) is bijective then \(\card{B} = \card{f\inv(B)}\text{.}\)