Chapter 6

Prove that if \(g \circ f\) is injective, then \(f\) is injective.

The members of the UN Peace Committee must choose, from among themselves, a presiding officer of their committee. For each member \(x\), let \(f(x)\) designate that member's choice for officer. If no two members vote alike, what is the range of \(f ?\)

Each of the following functions \(f\) is bijective. Describe its inverse. \(f:(0, \infty) \rightarrow(0, \infty)\), defined by \(f(x)=1 / x\)

\(f: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(f(x)=\sin x\) \(g: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(g(x)=e^{x}\). Find \(f \circ g\) and \(g \circ f\)

Determine whether each of the following functions is or is not \((a)\) injective, and \((b)\) surjective. \(f: \mathbb{R} \rightarrow(0, \infty)\), defined by \(f(x)=e^{x}\)

Proof By exhibiting a counterexample: \(-1\) is not equal to \(f(x)\) for any \(x \in \mathbb{R}\). \(f(x)=3 x+4\)

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

Each of the following functions \(f\) is bijective. Describe its inverse. \(f: \mathbb{R} \rightarrow(0, \infty)\), defined by \(f(x)=e^{x}\)

\(A\) and \(B\) are sets; \(f: A \times B \rightarrow B \times A\) is given by \(f(x, y)=(y, x)\). \(g: B \times A \rightarrow B\) is given by \(g(y, x)=y\). Find \(g \circ f\).

Determine whether each of the following functions is or is not \((a)\) injective, \((b)\) surjective. \(f: A \times B \rightarrow B \times A\), defined by \(f(x, y)=(y, x)\).

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

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.)

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

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

\(f:(0,1) \rightarrow \mathbb{R}\) 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 \(x .\)

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

Let \(f: A \rightarrow B\) and \(g: B \rightarrow A\) be functions. Suppose that \(y=f(x)\) iff \(x=g(y)\). Prove that \(f\) is bijective, and \(g=f^{-1}\)

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?

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

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

If \(A\) has \(n\) elements, how many functions are there from \(A\) to \(A\) ? How many bijective functions are there from \(A\) to \(A\) ?

Each of the following functions \(f\) is bijective. Describe its inverse. \(A=\\{a, b, c, d\\}, B=\\{1,2,3,4\\}\), and \(f: A \rightarrow B\) is given by $$ f=\left(\begin{array}{llll} a & b & c & d \\ 3 & 1 & 2 & 4 \end{array}\right) $$

\(A=\\{a, b, c, d\\} ; f\) and \(g\) are functions from \(A\) to \(A\); in the tabular form described on page 55 , they are given by $$ f=\left(\begin{array}{llll} a & b & c & d \\ a & c & a & c \end{array}\right) \quad g=\left(\begin{array}{llll} a & b & c & d \\ b & a & b & a \end{array}\right) $$ Give \(f \circ g\) and \(g \circ f\) in the same tabular form.

Determine whether each of the following functions is or is not \((a)\) injective, \((b)\) surjective. \(G\) is a group and \(f: G \rightarrow G\) is defined by \(f(x)=x^{-1}\).

Find a bijective function \(f\) from the set \(\mathbb{Z}\) of the integers to the set \(E\) of the even integers.

Proof By exhibiting a counterexample: \(-1\) is not equal to \(f(x)\) for any \(x \in \mathbb{R}\). \(f(x)=\left\\{\begin{array}{c}x \text { if } x \text { is rational } \\ 2 x \text { if } x \text { is irrational }\end{array}\right.\)

Each of the following functions \(f\) is bijective. Describe its inverse. \(G\) is a group, \(a \in G\), and \(f: G \rightarrow G\) is defined by \(f(x)=a x\).

\(G\) is a group, and \(a\) and \(b\) are elements of \(G\). \(f: G \rightarrow G\) is defined by \(f(x)=a x\) \(g: G \rightarrow G\) is defined by \(g(x)=b x\) Find \(f \circ g\) and \(g \circ f\)