Chapter 5, Functions Video Solutions, How to Prove It: A Structured Approach

Problem 1

(a) Let $A=\{1,2,3\}, B=\{4\}, f=\{(1,4),(2,4),(3,4)\}$. Is $f$ a function from $A$ to $B$ ?
(b) Let $A=\{1\}, B=\{2,3,4\}, f=\{(1,2),(1,3),(1,4)\}$. Is $f$ a function from $A$ to $B$ ?
(c) Let $f$ be the relation represented by the following graph. Is $f$ a function from $A$ to $B$ ?
(Figure can't copy)
(d) Let $C$ be the set of all cars registered in your state, and let $S$ be the set of all sequences of at most 10 letters and digits. Let $L=\{(c, s) \in$ $C \times S \mid$ the license plate number of the car $c$ is $s\}$. Is $L$ a function from $C$ to $S$ ?
(e) Let $W$ be the set of all words of English, and let $A$ be the set of all letters of the alphabet. Let $f=\{(w, a) \in W \times A \mid$ the letter $a$ occurs in the word $w\}$, and let $g=\{(w, a) \in W \times A \mid$ the letter $a$ is the first letter of the word $w\}$. Is $f$ a function from $W$ to $A$ ? How about $g$ ?
(f) John, Mary, Susan, and Fred go out to dinner and sit at a round table. Let $P=$ (John, Mary, Susan, Fred $\}$, and let $R=\{(p, q) \in P \times P$ ? the person $p$ is sitting immediately to the right of the person $q$}. Is $R$ a function from $P$ to $P$ ?

Wen Zheng

Numerade Educator

00:54

Problem 2

(a) Let $A=\{a, b, c\}, B=\{a, b\}$, and $f=\{(a, b),(b, b),(c, a)\}$. Then $f: A \rightarrow B$. What are $f(a), f(b)$, and $f(c)$ ?
(b) Let $N$ be the set of all countries and $C$ the set of cities. Let $H$ : $N \rightarrow C$ be the function defined by the rule that for every country $n$, $H(n)=$ the capital of the country $n$. What is $H$ (Italy)?
(c) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function defined by the formula
$$
f(x)=x^2-2 x .
$$
What is $f(2)$ ?
(d) Let $A=\{1,2,3\}$ and $B=\mathscr{P}(A)$. Let $F: B \rightarrow B$ be the function defined by the formula
$$
F(X)=A \backslash X
$$
What is $F(\{1,3\})$ ?

Ankit Gupta

Numerade Educator

03:22

Problem 3

Let $L$ be the function defined in part 3 of Example 5.1.2 and $H$ the function defined in exercise 2(b). Describe $L \circ H$ and $H \circ L$.

Ahmad Reda

Numerade Educator

01:21

Problem 4

Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$ defined by the following formulas:
$$
f(x)=\frac{1}{x^2+2}, \quad g(x)=2 x-1 .
$$
Find formulas for $(f \circ g)(x)$ and $(g \circ f)(x)$.

James Chok

Numerade Educator

08:35

Problem 5

Suppose $f: A \rightarrow B$ and $C \subseteq A$. The set $f \cap(C \times B)$, which is a relation from $C$ to $B$, is called the restriction of $f$ to $C$, and is sometimes denoted $f \nmid C$. In other words,
$$
f \mid C=f \cap(C \times B)
$$
(a) Prove that $f \mid C$ is a function from $C$ to $B$ and that for all $c \in$ $C, f(c)=(f \backslash C)(c)$.
(b) Suppose $g: C \rightarrow B$. Prove that $g=f \backslash C$ iff $g \subseteq f$.
(c) Let $g$ and $h$ be the functions defined in parts 2 and 3 of Example 5.1.3. Show that $g=h \downarrow \mathbb{Z}$.

Mengchun Cai

Numerade Educator

05:27

Problem 6

Suppose $A$ is a set. Show that $i_A$ is the only relation on $A$ that is both an equivalence relation on $A$ and also a function from $A$ to $A$.

Prathan Jarupoonphol

Numerade Educator

12:54

Problem 7

(a) Suppose $g: A \rightarrow B$ and let $R=\{(x, y) \in A \times A \mid g(x)=g(y)\}$. Show that $R$ is an equivalence relation on $A$.
(b) Suppose $R$ is an equivalence relation on $A$ and let $g: A \rightarrow A / R$ be the function defined by the formula $g(x)=[x]_R$. Show that $R=$ $\{(x, y) \in A \times A \mid g(x)=g(y)\}$.

Chris Trentman

Numerade Educator

12:54

Problem 8

Suppose $f: A \rightarrow B$ and $R$ is an equivalence relation on $A$. We will say that $f$ is compatible with $R$ if $\forall x \in A \forall y \in A(x R y \rightarrow f(x)=f(y))$. Let $g: A \rightarrow A / R$ be defined by the formula $g(x)=[x]_R$, as in part (b) of exercise 7.
(a) Suppose that there is a function $h: A / R \rightarrow B$ such that $h \circ g=f$. Prove that $f$ is compatible with $R$.
(b) Suppose that $f$ is compatible with $R$. Prove that there is a unique function $h: A / R \rightarrow B$ such that $h \circ g=f$.

Chris Trentman

Numerade Educator

07:56

Problem 9

Let $R=\{(x, y) \in \mathbb{Z} \times \mathbb{Z} \mid x \equiv y \bmod 5\}$. Recall that we saw in Section 4.6 that $R$ is an equivalence relation on $\mathbb{Z}$.
(a) Show that there is a unique function $h: \mathbb{Z} / R \rightarrow \mathbb{Z} / R$ such that for every integer $x, h\left([x]_R\right)=\left[x^2\right]_R$. (Hint: Use exercise 8.)
(b) Show that there is no function $h: \mathbb{Z} / R \rightarrow \mathbb{Z} / R$ such that for every integer $x, h\left([x]_R\right)=\left[2^x\right]_R$.

Anurag Kumar

Numerade Educator