91Ó°ÊÓ

Let \(A\) be a finite set. Explain why any injective function \(f: A \rightarrow A\) is necessarily surjective. (Look at part \(1 .\) )

Parts 1 and 2 , together, tell us that if \(g \circ f\) is bijective, then \(f\) is injective and \(g\) is surjective. Is the converse of this statement true: If \(f\) is injective and \(g\) surjective, is \(g=f\) bijective? (If "yes," prove it; if "no," give a counterexample.)

\(f:(0,1) \rightarrow \mathbb{R}\) is defined by \(f(x)=1 / x\). \(g: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(g(x)=\ln x\). Find \(g \circ f\).

Determine whether each of the following functions is or is not \((a)\) injective, and \((b)\) surjective. \(f: \mathbb{R} \rightarrow \mathbb{Z}\), defined by \(f(x)=\) the least integer greater than or equal to \(x .\)

Each of the following functions \(f\) is bijective. Describe its inverse. \(f: \mathbb{R} \rightarrow \mathbb{R}\), defined by \(f(x)=x^{3}+1\)

If \(A\) is a finite set, explain why any surjective function \(f: A \rightarrow A\) is necessarily injective.

Proof By exhibiting a counterexample: \(-1\) is not equal to \(f(x)\) for any \(x \in \mathbb{R}\). \(f(x)=|x|\)

Determine whether each of the following functions is or is not \((a)\) injective, and \((b)\) surjective. $$ f: \mathbb{Z} \rightarrow \mathbb{Z}, \text { defined by } f(n)= \begin{cases}n+1 & \text { if } n \text { is even } \\ n-1 & \text { if } n \text { is odd }\end{cases} $$

In school, Jack and Sam exchanged notes in a code \(f\) which consisted of spelling every word backwards and interchanging every letter \(s\) with t. Alternatively, they used a code \(g\) which interchanged the letters a with o, i with \(u, e\) with \(y\), and \(s\) with \(t\). Describe the codes \(f \circ g\) and \(g \circ f\). Are they the same?

Proof By exhibiting a counterexample: \(-1\) is not equal to \(f(x)\) for any \(x \in \mathbb{R}\). \(f(x)=x^{3}-3 x\)