SB Hints and Answers to Selected Even Exercises

Appendix C Hints and Answers to Selected Even Exercises

1 Preliminaries
1.4 Exercises

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)};

(d)

A \times D = \emptyset .

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1.4.6.

Hint.

x \in A \cup (B \cap C),

then either

x \in A

x \in B \cap C .

Thus,

x \in A \cup B

and

A \cup C .

Hence,

x \in (A \cup B) \cap (A \cup C) .

Therefore,

A \cup (B \cap C) \subset (A \cup B) \cap (A \cup C) .

Conversely, if

x \in (A \cup B) \cap (A \cup C),

then

x \in A \cup B

and

A \cup C .

Thus,

x \in A

x

is in both

B

and

C .

x \in A \cup (B \cap C)

and therefore

(A \cup B) \cap (A \cup C) \subset A \cup (B \cap C) .

Hence,

A \cup (B \cap C) = (A \cup B) \cap (A \cup C) .

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1.4.10.

Hint.

(A \cap B) \cup (A ∖ B) \cup (B ∖ 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 .

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1.4.14.

Hint.

A ∖ (B \cup C) = A \cap (B \cup C)^{'} = (A \cap A) \cap (B^{'} \cap C^{'}) = (A \cap B^{'}) \cap (A \cap C^{'}) = (A ∖ B) \cap (A ∖ C) .

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More Exercises

1.4.18.

Hint.

(a)

f

is one-to-one but not onto.

f (R) = {x \in R : x > 0} .

(c)

f

is neither one-to-one nor onto.

f (R) = {x : - 1 \leq x \leq 1} .

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1.4.20.

Hint.

(a)

f (n) = n + 1 .

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1.4.22.

Hint.

(a) Let

x, y \in A .

Then

g (f (x)) = (g \circ f) (x) = (g \circ f) (y) = g (f (y)) .

Thus,

f (x) = f (y)

and

x = y,

g \circ f

is one-to-one. (b) Let

c \in C,

then

c = (g \circ f) (x) = g (f (x))

for some

x \in A .

Since

f (x) \in B,

g

is onto.

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1.4.24.

Hint.

(a) Let

y \in f (A_{1} \cup A_{2}) .

Then there exists an

x \in A_{1} \cup A_{2}

such that

f (x) = y .

Hence,

y \in f (A_{1})

f (A_{2}) .

Therefore,

y \in f (A_{1}) \cup f (A_{2}) .

Consequently,

f (A_{1} \cup A_{2}) \subset f (A_{1}) \cup f (A_{2}) .

Conversely, if

y \in f (A_{1}) \cup f (A_{2}),

then

y \in f (A_{1})

f (A_{2}) .

Hence, there exists an

x \in A_{1}

or there exists an

x \in A_{2}

such that

f (x) = y .

Thus, there exists an

x \in A_{1} \cup A_{2}

such that

f (x) = y .

Therefore,

f (A_{1}) \cup f (A_{2}) \subset f (A_{1} \cup A_{2}),

and

f (A_{1} \cup A_{2}) = f (A_{1}) \cup f (A_{2}) .

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1.4.28.

Hint.

Let

X = N \cup {\sqrt{2}}

and define

x \sim y

x + y \in N .

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3 Groups
3.5 Exercises

3.5.2.

Hint.

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

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3.5.6.

Hint.

\begin{array}{ccccc} \cdot & 1 & 5 & 7 & 11 \\ 1 & 1 & 5 & 7 & 11 \\ 5 & 5 & 1 & 11 & 7 \\ 7 & 7 & 11 & 1 & 5 \\ 11 & 11 & 7 & 5 & 1 \end{array}

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3.5.8.

Hint.

Pick two matrices. Almost any pair will work.

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3.5.16.

Hint.

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

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3.5.18.

Hint.

Let

σ = (\begin{matrix} 1 & 2 & \dots & n \\ a_{1} & a_{2} & \dots & a_{n} \end{matrix})

be in

S_{n} .

All of the

a_{i}

s must be distinct. There are

n

ways to choose

a_{1},

n - 1

ways to choose

a_{2},

\dots,

2 ways to choose

a_{n - 1},

and only one way to choose

a_{n} .

Therefore, we can form

σ

n (n - 1) \dots 2 \cdot 1 = n!

ways.

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3.5.46.

Hint.

Look at

S_{3} .

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3.5.56.

Answer.

2

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3.5.58.

Answer.

n + 1

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3.5.60.

3.5.60.a

Answer.

4

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3.5.60.b

3.5.60.b.i

Answer.

8

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3.5.60.b.ii

Answer.

12

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5 Runestone Testing
5.9 Multiple Choice Exercises

5.9.4. Multiple-Choice, Randomized, One Answer.

Hint 1.

What did you see last time you went driving?

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Hint 2.

Maybe go out for a drive?

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5.9.6. Mathematical Multiple-Choice, Not Randomized, Multiple Answers.

Hint.

You can take a derivative on any one of the choices to see if it is correct or not, rather than using techniques of integration to find a single correct answer.

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5.10 Parsons Exercises

5.10.6. Parsons Problem, Mathematical Proof, Numbered Blocks.

Hint.

Dorothy will not be much help with this proof.

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5.18 Fill-In Exercises

5.18.6. Fill-In, New Markup Strings.

Hint.

Do you really need a hint? Carefully reread the question.

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5.18.10. Fill-In, Dynamic Math with Simple Numerical Answer.

5.18.12. Fill-In, Dynamic Math with Interdependent Formula Checking.

5.19 Hodgepodge

5.19.2. With Tasks in an Exercises Division.

5.19.2.a True/False.

Hint.

P_{n},

the vector space of polynomials with degree at most

n,

has dimension

n + 1

by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter

n ?

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5.20 Exercises that are Timed

Timed Exercises

5.20.2. 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?

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Appendix C Hints and Answers to Selected Even Exercises

1 Preliminaries1.4 Exercises

Warm-up

1.4.2.

1.4.6.

1.4.10.

1.4.14.

More Exercises

1.4.18.

1.4.20.

1.4.22.

1.4.24.

1.4.28.

3 Groups3.5 Exercises

3.5.2.

3.5.6.

3.5.8.

3.5.16.

3.5.18.

3.5.46.

3.5.56.

3.5.58.

3.5.60.

3.5.60.a

3.5.60.b

3.5.60.b.i

3.5.60.b.ii

5 Runestone Testing5.9 Multiple Choice Exercises

5.9.4. Multiple-Choice, Randomized, One Answer.

5.9.6. Mathematical Multiple-Choice, Not Randomized, Multiple Answers.

5.10 Parsons Exercises

5.10.6. Parsons Problem, Mathematical Proof, Numbered Blocks.

5.18 Fill-In Exercises

5.18.6. Fill-In, New Markup Strings.

5.18.10. Fill-In, Dynamic Math with Simple Numerical Answer.

5.18.12. Fill-In, Dynamic Math with Interdependent Formula Checking.

5.19 Hodgepodge

5.19.2. With Tasks in an Exercises Division.

5.19.2.a True/False.

5.20 Exercises that are Timed

Timed Exercises

5.20.2. Multiple-Choice, Not Randomized, One Answer.

1 Preliminaries
1.4 Exercises

3 Groups
3.5 Exercises

5 Runestone Testing
5.9 Multiple Choice Exercises