If (|A|=3) and (|B|=5), what are the numbers of all relations and all functions from (A) to (B), respectively?

Correct answer: (2^{15}) and (5^3)

The direct answer is A: there are \(2^{15}\) relations and \(5^3\) functions. A relation from \(A\) to \(B\) is any subset of the Cartesian product \(A\times B\). Since \(|A|=3\) and \(|B|=5\), the product has \(3\times5=15\) ordered pairs. Each pair may independently be included or excluded, giving \(2^{15}\) relations. A function is more restricted: each of the 3 domain elements must choose exactly one of the 5 codomain elements. Thus the number of functions is \(5\times5\times5=5^3\). Option A is correct. Option B gives 15, which is only the number of possible ordered pairs, not all subsets. Option C reverses the two counts. Option D uses incorrect sizes and formulas. Notice that every function is a relation, but most relations are not functions because they may omit an input or give one input several outputs. Memory cue: relations use \(2^{|A||B|}\); functions use \(|B|^{|A|}\).