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AppendixBHints and Solutions to Selected Exercises

Exercises1.3Exercises

Exercise1

Suppose that

\begin{align*} A & = \{ x : x \in \mathbb N \text{ and } x \text{ is even} \},\\ B & = \{x : x \in \mathbb N \text{ and } x \text{ is prime}\},\\ C & = \{ x : x \in \mathbb N \text{ and } x \text{ is a multiple of 5}\}. \end{align*}

Describe each of the following sets.

  1. \(A \cap B\)

  2. \(B \cap C\)

  3. \(A \cup B\)

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

Hint
Exercise2

If \(A = \{ a, b, c \}\text{,}\) \(B = \{ 1, 2, 3 \}\text{,}\) \(C = \{ x \}\text{,}\) and \(D = \emptyset\text{,}\) list all of the elements in each of the following sets.

  1. \(A \times B\)

  2. \(B \times A\)

  3. \(A \times B \times C\)

  4. \(A \times D\)

Hint
Exercise6

Prove \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\text{.}\)

Hint
Exercise10

Prove \(A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A)\text{.}\)

Hint
Exercise14

Prove \(A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)\text{.}\)

Hint
Exercise17

Which of the following relations \(f: {\mathbb Q} \rightarrow {\mathbb Q}\) define a mapping? In each case, supply a reason why \(f\) is or is not a mapping.

  1. \(\displaystyle f(p/q) = \frac{p+ 1}{p - 2}\)

  2. \(\displaystyle f(p/q) = \frac{3p}{3q}\)

  3. \(\displaystyle f(p/q) = \frac{p+q}{q^2}\)

  4. \(\displaystyle f(p/q) = \frac{3 p^2}{7 q^2} - \frac{p}{q}\)

Hint
Exercise18

Determine which of the following functions are one-to-one and which are onto. If the function is not onto, determine its range.

  1. \(f: {\mathbb R} \rightarrow {\mathbb R}\) defined by \(f(x) = e^x\)

  2. \(f: {\mathbb Z} \rightarrow {\mathbb Z}\) defined by \(f(n) = n^2 + 3\)

  3. \(f: {\mathbb R} \rightarrow {\mathbb R}\) defined by \(f(x) = \sin x\)

  4. \(f: {\mathbb Z} \rightarrow {\mathbb Z}\) defined by \(f(x) = x^2\)

Hint
Exercise20
  1. Define a function \(f: {\mathbb N} \rightarrow {\mathbb N}\) that is one-to-one but not onto.

  2. Define a function \(f: {\mathbb N} \rightarrow {\mathbb N}\) that is onto but not one-to-one.

Hint
Exercise22

Let \(f : A \rightarrow B\) and \(g : B \rightarrow C\) be maps.

  1. If \(f\) and \(g\) are both one-to-one functions, show that \(g \circ f\) is one-to-one.

  2. If \(g \circ f\) is onto, show that \(g\) is onto.

  3. If \(g \circ f\) is one-to-one, show that \(f\) is one-to-one.

  4. If \(g \circ f\) is one-to-one and \(f\) is onto, show that \(g\) is one-to-one.

  5. If \(g \circ f\) is onto and \(g\) is one-to-one, show that \(f\) is onto.

Hint
Exercise23

Define a function on the real numbers by

\begin{equation*} f(x) = \frac{x + 1}{x - 1}. \end{equation*}

What are the domain and range of \(f\text{?}\) What is the inverse of \(f\text{?}\) Compute \(f \circ f^{-1}\) and \(f^{-1} \circ f\text{.}\)

Hint
Exercise24

Let \(f: X \rightarrow Y\) be a map with \(A_1, A_2 \subset X\) and \(B_1, B_2 \subset Y\text{.}\)

  1. Prove \(f( A_1 \cup A_2 ) = f( A_1) \cup f( A_2 )\text{.}\)

  2. Prove \(f( A_1 \cap A_2 ) \subset f( A_1) \cap f( A_2 )\text{.}\) Give an example in which equality fails.

  3. Prove \(f^{-1}( B_1 \cup B_2 ) = f^{-1}( B_1) \cup f^{-1}(B_2 )\text{,}\) where

    \begin{equation*} f^{-1}(B) = \{ x \in X : f(x) \in B \}. \end{equation*}
  4. Prove \(f^{-1}( B_1 \cap B_2 ) = f^{-1}( B_1) \cap f^{-1}( B_2 )\text{.}\)

  5. Prove \(f^{-1}( Y \setminus B_1 ) = X \setminus f^{-1}( B_1)\text{.}\)

Hint
Exercise25

Determine whether or not the following relations are equivalence relations on the given set. If the relation is an equivalence relation, describe the partition given by it. If the relation is not an equivalence relation, state why it fails to be one.

  1. \(x \sim y\) in \({\mathbb R}\) if \(x \geq y\)

  2. \(m \sim n\) in \({\mathbb Z}\) if \(mn > 0\)

  3. \(x \sim y\) in \({\mathbb R}\) if \(|x - y| \leq 4\)

  4. \(m \sim n\) in \({\mathbb Z}\) if \(m \equiv n \pmod{6}\)

Hint
Exercise28

Find the error in the following argument by providing a counterexample. “The reflexive property is redundant in the axioms for an equivalence relation. If \(x \sim y\text{,}\) then \(y \sim x\) by the symmetric property. Using the transitive property, we can deduce that \(x \sim x\text{.}\)”

Hint

Exercises2.3Exercises

Exercise1

Prove that

\begin{equation*} 1^2 + 2^2 + \cdots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \end{equation*}

for \(n \in {\mathbb N}\text{.}\)

Hint
Exercise3

Prove that \(n! \gt 2^n\) for \(n \geq 4\text{.}\)

Hint
Exercise8

Prove the Leibniz rule for \(f^{(n)} (x)\text{,}\) where \(f^{(n)}\) is the \(n\)th derivative of \(f\text{;}\) that is, show that

\begin{equation*} (fg)^{(n)}(x) = \sum_{k = 0}^{n} \binom{n}{k} f^{(k)}(x) g^{(n - k)}(x). \end{equation*} Hint
Exercise11

If \(x\) is a nonnegative real number, then show that \((1 + x)^n - 1 \geq nx\) for \(n = 0, 1, 2, \ldots\text{.}\)

Hint
Exercise17Fibonacci Numbers

The Fibonacci numbers are

\begin{equation*} 1, 1, 2, 3, 5, 8, 13, 21, \ldots. \end{equation*}

We can define them inductively by \(f_1 = 1\text{,}\) \(f_2 = 1\text{,}\) and \(f_{n + 2} = f_{n + 1} + f_n\) for \(n \in {\mathbb N}\text{.}\)

  1. Prove that \(f_n \lt 2^n\text{.}\)

  2. Prove that \(f_{n + 1} f_{n - 1} = f^2_n + (-1)^n\text{,}\) \(n \geq 2\text{.}\)

  3. Prove that \(f_n = [(1 + \sqrt{5}\, )^n - (1 - \sqrt{5}\, )^n]/ 2^n \sqrt{5}\text{.}\)

  4. Show that \(\lim_{n \rightarrow \infty} f_n / f_{n + 1} = (\sqrt{5} - 1)/2\text{.}\)

  5. Prove that \(f_n\) and \(f_{n + 1}\) are relatively prime.

Hint
Exercise19

Let \(x, y \in {\mathbb N}\) be relatively prime. If \(xy\) is a perfect square, prove that \(x\) and \(y\) must both be perfect squares.

Hint
Exercise23

Define the least common multiple of two nonzero integers \(a\) and \(b\text{,}\) denoted by \(\lcm(a,b)\text{,}\) to be the nonnegative integer \(m\) such that both \(a\) and \(b\) divide \(m\text{,}\) and if \(a\) and \(b\) divide any other integer \(n\text{,}\) then \(m\) also divides \(n\text{.}\) Prove that any two integers \(a\) and \(b\) have a unique least common multiple.

Hint
Exercise27

Let \(a, b, c \in {\mathbb Z}\text{.}\) Prove that if \(\gcd(a,b) = 1\) and \(a \mid bc\text{,}\) then \(a \mid c\text{.}\)

Hint
Exercise29

Prove that there are an infinite number of primes of the form \(6n + 5\text{.}\)

Hint

Exercises3.4Exercises

Exercise1

Find all \(x \in {\mathbb Z}\) satisfying each of the following equations.

  1. \(3x \equiv 2 \pmod{7}\)

  2. \(5x + 1 \equiv 13 \pmod{23}\)

  3. \(5x + 1 \equiv 13 \pmod{26}\)

  4. \(9x \equiv 3 \pmod{5}\)

  5. \(5x \equiv 1 \pmod{6}\)

  6. \(3x \equiv 1 \pmod{6}\)

Hint
Exercise2

Which of the following multiplication tables defined on the set \(G = \{ a, b, c, d \}\) form a group? Support your answer in each case.

  1. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & c & d & a \\ b & b & b & c & d \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
  2. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & d & c \\ c & c & d & a & b \\ d & d & c & b & a \end{array} \end{equation*}
  3. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & c & d & a \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
  4. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & c & d \\ c & c & b & a & d \\ d & d & d & b & c \end{array} \end{equation*}
Hint
Exercise6

Give a multiplication table for the group \(U(12)\text{.}\)

Hint
Exercise8

Give an example of two elements \(A\) and \(B\) in \(GL_2({\mathbb R})\) with \(AB \neq BA\text{.}\)

Hint
Exercise15

Prove or disprove that every group containing six elements is abelian.

Hint
Exercise16

Give a specific example of some group \(G\) and elements \(g, h \in G\) where \((gh)^n \neq g^nh^n\text{.}\)

Hint
Exercise17

Give an example of three different groups with eight elements. Why are the groups different?

Hint
Exercise18

Show that there are \(n!\) permutations of a set containing \(n\) items.

Hint
Exercise25

Let \(a\) and \(b\) be elements in a group \(G\text{.}\) Prove that \(ab^na^{-1} = (aba^{-1})^n\) for \(n \in \mathbb Z\text{.}\)

Hint
Exercise31

Show that if \(a^2 = e\) for all elements \(a\) in a group \(G\text{,}\) then \(G\) must be abelian.

Hint
Exercise35

Find all the subgroups of the symmetry group of an equilateral triangle.

Hint
Exercise41

Prove that

\begin{equation*} G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \} \end{equation*}

is a subgroup of \({\mathbb R}^{\ast}\) under the group operation of multiplication.

Hint
Exercise46

Prove or disprove: If \(H\) and \(K\) are subgroups of a group \(G\text{,}\) then \(H \cup K\) is a subgroup of \(G\text{.}\)

Hint
Exercise49

Let \(a\) and \(b\) be elements of a group \(G\text{.}\) If \(a^4b = ba\) and \(a^3 = e\text{,}\) prove that \(ab = ba\text{.}\)

Hint

Exercises4.5Exercises

Exercise1

Prove or disprove each of the following statements.

  1. All of the generators of \({\mathbb Z}_{60}\) are prime.

  2. \(U(8)\) is cyclic.

  3. \({\mathbb Q}\) is cyclic.

  4. If every proper subgroup of a group \(G\) is cyclic, then \(G\) is a cyclic group.

  5. A group with a finite number of subgroups is finite.

Hint
Exercise2

Find the order of each of the following elements.

  1. \(5 \in {\mathbb Z}_{12}\)

  2. \(\sqrt{3} \in {\mathbb R}\)

  3. \(\sqrt{3} \in {\mathbb R}^\ast\)

  4. \(-i \in {\mathbb C}^\ast\)

  5. 72 in \({\mathbb Z}_{240}\)

  6. 312 in \({\mathbb Z}_{471}\)

Hint
Exercise3

List all of the elements in each of the following subgroups.

  1. The subgroup of \({\mathbb Z}\) generated by 7

  2. The subgroup of \({\mathbb Z}_{24}\) generated by 15

  3. All subgroups of \({\mathbb Z}_{12}\)

  4. All subgroups of \({\mathbb Z}_{60}\)

  5. All subgroups of \({\mathbb Z}_{13}\)

  6. All subgroups of \({\mathbb Z}_{48}\)

  7. The subgroup generated by 3 in \(U(20)\)

  8. The subgroup generated by 5 in \(U(18)\)

  9. The subgroup of \({\mathbb R}^\ast\) generated by 7

  10. The subgroup of \({\mathbb C}^\ast\) generated by \(i\) where \(i^2 = -1\)

  11. The subgroup of \({\mathbb C}^\ast\) generated by \(2i\)

  12. The subgroup of \({\mathbb C}^\ast\) generated by \((1 + i) / \sqrt{2}\)

  13. The subgroup of \({\mathbb C}^\ast\) generated by \((1 + \sqrt{3}\, i) / 2\)

Hint
Exercise4

Find the subgroups of \(GL_2( {\mathbb R })\) generated by each of the following matrices.

  1. \(\displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)

  2. \(\displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}\)

  3. \(\displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}\)

  4. \(\displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}\)

  5. \(\displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}\)

  6. \(\displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}\)

Hint
Exercise10

Find all elements of finite order in each of the following groups. Here the “\(\ast\)” indicates the set with zero removed.

  1. \({\mathbb Z}\)

  2. \({\mathbb Q}^\ast\)

  3. \({\mathbb R}^\ast\)

Hint
Exercise11

If \(a^{24} =e\) in a group \(G\text{,}\) what are the possible orders of \(a\text{?}\)

Hint
Exercise15

Evaluate each of the following.

  1. \((3-2i)+ (5i-6)\)

  2. \((4-5i)-\overline{(4i -4)}\)

  3. \((5-4i)(7+2i)\)

  4. \((9-i) \overline{(9-i)}\)

  5. \(i^{45}\)

  6. \((1+i)+\overline{(1+i)}\)

Hint
Exercise16

Convert the following complex numbers to the form \(a + bi\text{.}\)

  1. \(2 \cis(\pi / 6 )\)

  2. \(5 \cis(9\pi/4)\)

  3. \(3 \cis(\pi)\)

  4. \(\cis(7\pi/4) /2\)

Hint
Exercise17

Change the following complex numbers to polar representation.

  1. \(1-i\)

  2. \(-5\)

  3. \(2+2i\)

  4. \(\sqrt{3} + i\)

  5. \(-3i\)

  6. \(2i + 2 \sqrt{3}\)

Hint
Exercise18

Calculate each of the following expressions.

  1. \((1+i)^{-1}\)

  2. \((1 - i)^{6}\)

  3. \((\sqrt{3} + i)^{5}\)

  4. \((-i)^{10}\)

  5. \(((1-i)/2)^{4}\)

  6. \((-\sqrt{2} - \sqrt{2}\, i)^{12}\)

  7. \((-2 + 2i)^{-5}\)

Hint
Exercise22

Calculate each of the following.

  1. \(292^{3171} \pmod{ 582}\)

  2. \(2557^{ 341} \pmod{ 5681}\)

  3. \(2071^{ 9521} \pmod{ 4724}\)

  4. \(971^{ 321} \pmod{ 765}\)

Hint
Exercise27

If \(g\) and \(h\) have orders 15 and 16 respectively in a group \(G\text{,}\) what is the order of \(\langle g \rangle \cap \langle h \rangle \text{?}\)

Hint
Exercise31

Let \(G\) be an abelian group. Show that the elements of finite order in \(G\) form a subgroup. This subgroup is called the torsion subgroup of \(G\text{.}\)

Hint
Exercise37

Prove that if \(G\) has no proper nontrivial subgroups, then \(G\) is a cyclic group.

Hint