Problem 17
Explain why a function from an \(n\)-element set to an \(n\)-element set is one- to-one if and only if it is onto.
Problem 18
The function \(g\) is called an inverse to the function \(f\) if the domain of \(g\) is the range of \(f\), if \(g(f(x))=x\) for every \(x\) in the domain of \(f\), and if \(f(g(y))=y\) for each \(y\) in the range of \(f\). a. Explain why a function is a bijection if and only if it has an inverse function. b. Explain why a function that has an inverse function has only one inverse function.
Problem 18
Apply calculus and the binomial theorem to \((1+x)^{n}\) to show that $$ \left(\begin{array}{l} n \\ 1 \end{array}\right)+2\left(\begin{array}{l} n \\ 2 \end{array}\right)+3\left(\begin{array}{l} n \\ 3 \end{array}\right)+\cdots=n 2^{n-1} . $$
Problem 19
True or false: \(\left(\begin{array}{l}n \\\ k\end{array}\right)=\left(\begin{array}{c}n-2 \\\ k-2\end{array}\right)+\left(\begin{array}{c}n-2 \\\ k-1\end{array}\right)+\left(\begin{array}{c}n-2 \\ k\end{array}\right)\). If true, give a proof. If false, give values of \(n\) and \(k\) that show the statement is false, find an analogous true statement, and prove it.
