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PreTeXt Sample Book Abstract Algebra (SAMPLE ONLY)

Appendix D Hints and Answers to Selected Exercises

1 Preliminaries
1.4 Exercises

Warm-up

1.4.2.
Hint.
(a) AΓ—B={(a,1),(a,2),(a,3),(b,1),(b,2),(b,3),(c,1),(c,2),(c,3)}; (d) AΓ—D=βˆ….
1.4.6.
Hint.
If x∈Aβˆͺ(B∩C), then either x∈A or x∈B∩C. Thus, x∈AβˆͺB and AβˆͺC. Hence, x∈(AβˆͺB)∩(AβˆͺC). Therefore, Aβˆͺ(B∩C)βŠ‚(AβˆͺB)∩(AβˆͺC). Conversely, if x∈(AβˆͺB)∩(AβˆͺC), then x∈AβˆͺB and AβˆͺC. Thus, x∈A or x is in both B and C. So x∈Aβˆͺ(B∩C) and therefore (AβˆͺB)∩(AβˆͺC)βŠ‚Aβˆͺ(B∩C). Hence, Aβˆͺ(B∩C)=(AβˆͺB)∩(AβˆͺC).
1.4.10.
Hint.
(A∩B)βˆͺ(Aβˆ–B)βˆͺ(Bβˆ–A)=(A∩B)βˆͺ(A∩Bβ€²)βˆͺ(B∩Aβ€²)=[A∩(BβˆͺBβ€²)]βˆͺ(B∩Aβ€²)=Aβˆͺ(B∩Aβ€²)=(AβˆͺB)∩(AβˆͺAβ€²)=AβˆͺB.
1.4.14.
Hint.
Aβˆ–(BβˆͺC)=A∩(BβˆͺC)β€²=(A∩A)∩(Bβ€²βˆ©Cβ€²)=(A∩Bβ€²)∩(A∩Cβ€²)=(Aβˆ–B)∩(Aβˆ–C).

More Exercises

1.4.18.
Hint.
(a) f is one-to-one but not onto. f(R)={x∈R:x>0}. (c) f is neither one-to-one nor onto. f(R)={x:βˆ’1≀x≀1}.
1.4.22.
Hint.
(a) Let x,y∈A. Then g(f(x))=(g∘f)(x)=(g∘f)(y)=g(f(y)). Thus, f(x)=f(y) and x=y, so g∘f is one-to-one. (b) Let c∈C, then c=(g∘f)(x)=g(f(x)) for some x∈A. Since f(x)∈B, g is onto.
1.4.24.
Hint.
(a) Let y∈f(A1βˆͺA2). Then there exists an x∈A1βˆͺA2 such that f(x)=y. Hence, y∈f(A1) or f(A2). Therefore, y∈f(A1)βˆͺf(A2). Consequently, f(A1βˆͺA2)βŠ‚f(A1)βˆͺf(A2). Conversely, if y∈f(A1)βˆͺf(A2), then y∈f(A1) or f(A2). Hence, there exists an x∈A1 or there exists an x∈A2 such that f(x)=y. Thus, there exists an x∈A1βˆͺA2 such that f(x)=y. Therefore, f(A1)βˆͺf(A2)βŠ‚f(A1βˆͺA2), and f(A1βˆͺA2)=f(A1)βˆͺf(A2).
1.4.28.
Hint.
Let X=Nβˆͺ{2} and define x∼y if x+y∈N.

2 The Integers
2.4 Exercises

2.4.1.

Answer.
The base case, S(1):[1(1+1)(2(1)+1)]/6=1=12 is true.
Assume that S(k):12+22+β‹―+k2=[k(k+1)(2k+1)]/6 is true. Then
12+22+β‹―+k2+(k+1)2=[k(k+1)(2k+1)]/6+(k+1)2=[(k+1)((k+1)+1)(2(k+1)+1)]/6,
and so S(k+1) is true. Thus, S(n) is true for all positive integers n.

2.4.3.

Answer.
The base case, S(4):4!=24>16=24 is true. Assume S(k):k!>2k is true. Then (k+1)!=k!(k+1)>2kβ‹…2=2k+1, so S(k+1) is true. Thus, S(n) is true for all positive integers n.

2.4.11.

Hint.
The base case, S(0):(1+x)0βˆ’1=0β‰₯0=0β‹…x is true. Assume S(k):(1+x)kβˆ’1β‰₯kx is true. Then
(1+x)k+1βˆ’1=(1+x)(1+x)kβˆ’1=(1+x)k+x(1+x)kβˆ’1β‰₯kx+x(1+x)kβ‰₯kx+x=(k+1)x,
so S(k+1) is true. Therefore, S(n) is true for all positive integers n.

2.4.23.

Hint.
Let S={s∈N:a∣s, b∣s}. Then Sβ‰ βˆ…, since |ab|∈S. By the Principle of Well-Ordering, S contains a least element m. To show uniqueness, suppose that a∣n and b∣n for some n∈N. By the division algorithm, there exist unique integers q and r such that n=mq+r, where 0≀r<m. Since a and b divide both m, and n, it must be the case that a and b both divide r. Thus, r=0 by the minimality of m. Therefore, m∣n.

2.4.27.

Hint.
Since gcd(a,b)=1, there exist integers r and s such that ar+bs=1. Thus, acr+bcs=c. Since a divides both bc and itself, a must divide c.

2.4.29.

Hint.
Every prime must be of the form 2, 3, 6n+1, or 6n+5. Suppose there are only finitely many primes of the form 6k+5.

3 Groups
3.5 Exercises

3.5.1.

Hint.
(a) 3+7Z={…,βˆ’4,3,10,…}; (c) 18+26Z; (e) 5+6Z.

3.5.15.

Hint.
There is a nonabelian group containing six elements.

3.5.16.

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

3.5.18.

Hint.
Let
Οƒ=(12β‹―na1a2β‹―an)
be in Sn. All of the ais must be distinct. There are n ways to choose a1, nβˆ’1 ways to choose a2, …, 2 ways to choose anβˆ’1, and only one way to choose an. Therefore, we can form Οƒ in n(nβˆ’1)β‹―2β‹…1=n! ways.

3.5.25.

Hint.
(abaβˆ’1)n=(abaβˆ’1)(abaβˆ’1)β‹―(abaβˆ’1)=ab(aaβˆ’1)b(aaβˆ’1)bβ‹―b(aaβˆ’1)baβˆ’1=abnaβˆ’1.

3.5.31.

Hint.
Since abab=(ab)2=e=a2b2=aabb, we know that ba=ab.

3.5.35.

Hint.
H1={id}, H2={id,ρ1,ρ2}, H3={id,μ1}, H4={id,μ2}, H5={id,μ3}, S3.

3.5.41.

Hint.
The identity of G is 1=1+02. Since (a+b2)(c+d2)=(ac+2bd)+(ad+bc)2, G is closed under multiplication. Finally, (a+b2)βˆ’1=a/(a2βˆ’2b2)βˆ’b2/(a2βˆ’2b2).

3.5.49.

Hint.
Since a4b=ba, it must be the case that b=a6b=a2ba, and we can conclude that ab=a3ba=ba.

5 Runestone Testing
5.8 True/False Exercises

5.8.1. True/False.

Hint.
Pn, 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?

5.9 Multiple Choice Exercises

5.9.1. 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?

5.9.3. Multiple-Choice, Not Randomized, Multiple Answers.

Hint.
Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?

5.9.4. Multiple-Choice, Randomized, One Answer.

Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?

5.9.5. Multiple-Choice, Randomized, Multiple Answers.

Hint.
Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?

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.

5.9.7. 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?

5.10 Parsons Exercises

5.10.1. Parsons Problem, Mathematical Proof.

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

5.10.6. Parsons Problem, Mathematical Proof, Numbered Blocks.

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

5.12 Matching Exercises

5.12.3. Cardsort Problem, Linear Algebra.

Hint.
For openers, a basis for a subspace must be a subset of the subspace.

5.13 Clickable Area Exercises

5.13.3. Clickable Areas, Text in a Table.

Hint.
Python boolean variables begin with capital latters.

5.18 Fill-In Exercises

5.18.6. Fill-In, New Markup Strings.

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

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

5.18.11. Fill-In, Dynamic Math with Formulas as Answers.

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.
Pn, 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?

5.20 Exercises that are Timed

Timed Exercises

5.20.1. True/False.
Hint.
Pn, 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?
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?

5.27 Group Exercises

5.27.1. 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?