# PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

## AppendixCHints and Answers to Selected Even Exercises

### 1Preliminaries1.4Exercises

#### Warm-up

##### 1.4.2.
Hint.
(a) $$A \times B = \{ (a,1), (a,2), (a,3), (b,1), (b,2), (b,3), (c,1), (c,2), (c,3) \}\text{;}$$ (d) $$A \times D = \emptyset\text{.}$$
##### 1.4.6.
Hint.
If $$x \in A \cup (B \cap C)\text{,}$$ then either $$x \in A$$ or $$x \in B \cap C\text{.}$$ Thus, $$x \in A \cup B$$ and $$A \cup C\text{.}$$ Hence, $$x \in (A \cup B) \cap (A \cup C)\text{.}$$ Therefore, $$A \cup (B \cap C) \subset (A \cup B) \cap (A \cup C)\text{.}$$ Conversely, if $$x \in (A \cup B) \cap (A \cup C)\text{,}$$ then $$x \in A \cup B$$ and $$A \cup C\text{.}$$ Thus, $$x \in A$$ or $$x$$ is in both $$B$$ and $$C\text{.}$$ So $$x \in A \cup (B \cap C)$$ and therefore $$(A \cup B) \cap (A \cup C) \subset A \cup (B \cap C)\text{.}$$ Hence, $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\text{.}$$
##### 1.4.10.
Hint.
$$(A \cap B) \cup (A \setminus B) \cup (B \setminus A) = (A \cap B) \cup (A \cap B') \cup (B \cap A') = [A \cap (B \cup B')] \cup (B \cap A') = A \cup (B \cap A') = (A \cup B) \cap (A \cup A') = A \cup B\text{.}$$
##### 1.4.14.
Hint.
$$A \setminus (B \cup C) = A \cap (B \cup C)' = (A \cap A) \cap (B' \cap C') = (A \cap B') \cap (A \cap C') = (A \setminus B) \cap (A \setminus C)\text{.}$$

#### More Exercises

##### 1.4.18.
Hint.
(a) $$f$$ is one-to-one but not onto. $$f({\mathbb R} ) = \{ x \in {\mathbb R} : x \gt 0 \}\text{.}$$ (c) $$f$$ is neither one-to-one nor onto. $$f(\mathbb R) = \{ x : -1 \leq x \leq 1 \}\text{.}$$
##### 1.4.20.
Hint.
(a) $$f(n) = n + 1\text{.}$$
##### 1.4.22.
Hint.
(a) Let $$x, y \in A\text{.}$$ Then $$g(f(x)) = (g \circ f)(x) = (g \circ f)(y) = g(f(y))\text{.}$$ Thus, $$f(x) = f(y)$$ and $$x = y\text{,}$$ so $$g \circ f$$ is one-to-one. (b) Let $$c \in C\text{,}$$ then $$c = (g \circ f)(x) = g(f(x))$$ for some $$x \in A\text{.}$$ Since $$f(x) \in B\text{,}$$ $$g$$ is onto.
##### 1.4.24.
Hint.
(a) Let $$y \in f(A_1 \cup A_2)\text{.}$$ Then there exists an $$x \in A_1 \cup A_2$$ such that $$f(x) = y\text{.}$$ Hence, $$y \in f(A_1)$$ or $$f(A_2) \text{.}$$ Therefore, $$y \in f(A_1) \cup f(A_2)\text{.}$$ Consequently, $$f(A_1 \cup A_2) \subset f(A_1) \cup f(A_2)\text{.}$$ Conversely, if $$y \in f(A_1) \cup f(A_2)\text{,}$$ then $$y \in f(A_1)$$ or $$f(A_2)\text{.}$$ Hence, there exists an $$x \in A_1$$ or there exists an $$x \in A_2$$ such that $$f(x) = y\text{.}$$ Thus, there exists an $$x \in A_1 \cup A_2$$ such that $$f(x) = y\text{.}$$ Therefore, $$f(A_1) \cup f(A_2) \subset f(A_1 \cup A_2)\text{,}$$ and $$f(A_1 \cup A_2) = f(A_1) \cup f(A_2)\text{.}$$
##### 1.4.28.
Hint.
Let $$X = {\mathbb N} \cup \{ \sqrt{2}\, \}$$ and define $$x \sim y$$ if $$x + y \in {\mathbb N}\text{.}$$

### 3Groups3.5Exercises

#### 3.5.2.

Hint.
(a) Not a group; (c) a group.

#### 3.5.6.

Hint.
\begin{equation*} \begin{array}{c|cccc} \cdot & 1 & 5 & 7 & 11 \\ \hline 1 & 1 & 5 & 7 & 11 \\ 5 & 5 & 1 & 11 & 7 \\ 7 & 7 & 11 & 1 & 5 \\ 11 & 11 & 7 & 5 & 1 \end{array} \end{equation*}

#### 3.5.8.

Hint.
Pick two matrices. Almost any pair will work.

#### 3.5.16.

Hint.
Look at the symmetry group of an equilateral triangle or a square.

#### 3.5.18.

Hint.
Let
\begin{equation*} \sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{pmatrix} \end{equation*}
be in $$S_n\text{.}$$ All of the $$a_i$$s must be distinct. There are $$n$$ ways to choose $$a_1\text{,}$$ $$n-1$$ ways to choose $$a_2\text{,}$$ $$\ldots\text{,}$$ 2 ways to choose $$a_{n - 1}\text{,}$$ and only one way to choose $$a_n\text{.}$$ Therefore, we can form $$\sigma$$ in $$n(n - 1) \cdots 2 \cdot 1 = n!$$ ways.

#### 3.5.46.

Hint.
Look at $$S_3\text{.}$$

#### 3.5.56.

$$2$$

#### 3.5.58.

$$n+1$$

#### 3.5.60.

##### 3.5.60.a
$$4$$
##### 3.5.60.b
###### 3.5.60.b.i
$$8$$
###### 3.5.60.b.ii
$$12$$

### 5Runestone Testing5.8Multiple Choice Exercises

#### 5.8.2.Multiple-Choice, Not Randomized, Multiple Answers.

Hint.
Do you know the acronymâ€¦ROY G BIV for the colors of a rainbow, and their order?

#### 5.8.4.Multiple-Choice, Randomized, Multiple Answers.

Hint.
Do you know the acronymâ€¦ROY G BIV for the colors of a rainbow, and their order?

### 5.9Parsons Exercises

#### 5.9.4.Parsons Problem, Mathematical Proof, Numbered Blocks.

Hint.
Dorothy will not be much help with this proof.

### 5.17Exercises that are Timed

#### Timed Exercises

##### 5.17.2.Multiple-Choice, Not Randomized, One Answer.
Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?