91Ó°ÊÓ

Suppose \(X\) is a set with \(m\) elements, and \(Y\) is a set with \(n\) elements. How many elements does \(X \times Y\) have? Is the answer the same if one or both of the sets is empty?

Let \(X\) and \(Y\) be finite non-empty sets, with \(m\) and \(n\) elements, respectively. How many functions are there from \(X\) to \(Y ?\) How many injections? How many surjections? How many bijections?

Let the real function \(f\) be strictly increasing. Show that for any \(b \in \mathbb{R}, f^{-1}(b)\) is either empty or consists of a single element, and that \(f\) is therefore an injection. If \(f\) is also a bijection, is the inverse function of \(f\) also strictly increasing?