HW2 詳解

Question 1

問題1

page 131, chapter 2.1 Exercises 2

Use set builder notation to give a description of each of these sets.
  1. $\{0, 3, 6, 9, 12\}$
  2. $\{-3,-2,-1, 0, 1, 2, 3\}$
  3. $\{m, n, o, p\}$

問題1

page 131, chapter 2.1 Exercises 2

Use set builder notation to give a description of each of these sets.
  1. $\{0, 3, 6, 9, 12\}$
  2. $\{-3,-2,-1, 0, 1, 2, 3\}$
  3. $\{m, n, o, p\}$

  1. $\{3n\ |\ n = 0, 1, 2, 3, 4\}$ or $\{x\ |\ x\ is\ a\ multiple\ of\ 3\wedge 0\le x\le 12\}$
  2. $\{x\ |\ -3\le x\le 3\}$, where we are assuming that the domain (universe of discourse) is the set of integers.
  3. $\{x\ |\ x\ is\ a\ letter\ of\ the\ word\ monopoly\ other\ than\ l\ or\ y\}$

Question 2

問題2

page 132, chapter 2.1 Exercises 11

Determine whether each of these statements is true or false.
  1. $0 \in \emptyset$
  2. $\emptyset \in \{0\}$
  3. $\{0\} \subset \emptyset$
  4. $\emptyset \subset \{0\}$
  5. $\{0\} \in \{0\}$
  6. $\{0\} \subset \{0\}$
  7. $\{\emptyset\} \subseteq \{\emptyset\}$
Determine whether each of these statements is true or false.
  1. $0 \in \emptyset$
  2. $\emptyset \in \{0\}$
  3. $\{0\} \subset \emptyset$
  4. $\emptyset \subset \{0\}$
  5. $\{0\} \in \{0\}$
  6. $\{0\} \subset \{0\}$
  7. $\{\emptyset\} \subseteq \{\emptyset\}$
$\emptyset = \{ \} \neq \{\emptyset\}$
$a \in A$ : a is an element of the set A.
$A \subseteq B$ : A is a subset of the set B, $\forall x(x \in A \rightarrow x \in B)$.
$A \subset B$ : A is a proper subset of B, $\forall x(x \in A \rightarrow x \in B) \wedge \exists x(x \in B \wedge x \not\in A)$.

  1. False
  2. False
  3. False
  4. True
  5. False
  6. False
  7. True

Question 3

問題3

page 132, Section 2.1 Exercise 36

Find $A^3$ if
  1. $A = \{a\}$
  2. $A = \{0,a\}$

問題3

page 132, Section 2.1 Exercise 36

Find $A^3$ if
  1. $A = \{a\}$
  2. $A = \{0,a\}$
$A^n = \{(a_1, a_2, … , a_n) ∣ a_i ∈ A \: for \: i = 1, 2, … , n\}$

  1. $\{(a, a, a)\}$
  2. $\{(0, 0, 0),(0, 0, a),(0, a, 0),(0, a, a),(a, 0, 0),(a, 0, a),(a, a, 0),(a, a, a)\}$

Question 4

問題4

page 144, Section 2.2 Exercise 4

Let $A = \{a, b, c, d, e\}$ and $B = \{a, b, c, d, e, f, g, h\}$. Find
  1. $A ∪ B$
  2. $A ∩ B$
  3. $A − B$
  4. $B − A$

問題4

page 144, Section 2.2 Exercise 4

Let $A = \{a, b, c, d, e\}$ and $B = \{a, b, c, d, e, f, g, h\}$. Find
  1. $A ∪ B$
  2. $A ∩ B$
  3. $A − B$
  4. $B − A$
$A = \{a, b, c, d, e\}$
$B = \{a, b, c, d, e, f, g, h\}$

  1. $\{a, b, c, d, e, f, g, h\}=B$
  2. $\{a, b, c, d, e\}=A$
  3. $\varnothing$
  4. $\{f, g, h\}$

Question 5

問題5

page 163, Section 2.3 Exercise 38

Find $f ◦g$ and $g◦f$ , where $f(x) = x^2 + 1$ and $g(x) = x + 2$, are functions from $\textbf{R}$ to $\textbf{R}$

問題5

page 163, Section 2.3 Exercise 38

Find $f ◦g$ and $g◦f$ , where $f(x) = x^2 + 1$ and $g(x) = x + 2$, are functions from $\textbf{R}$ to $\textbf{R}$
Given: $g: R \to R$ and $f: R \to R$

We have $(f ◦g)(x)=f(g(x))=f(x+2)=(x+2)^2+1=x^2+4x+5$,
whereas $(g◦f)(x) = g(f(x))=g(x^2+1)=x^2+1+2=x^2+3$. Note that they are not equal

Question 6

問題6

page 179, Section 2.4 Exercise 34

Compute each of these double sums.
  1. $\sum\limits_{i=1}^3 \sum\limits_{j=1}^2 (i-j)$
  2. $\sum\limits_{i=0}^3 \sum\limits_{j=0}^2 (3i+2j)$
  3. $\sum\limits_{i=1}^3 \sum\limits_{j=0}^2 j$
  4. $\sum\limits_{i=1}^2 \sum\limits_{j=0}^3 (i^2j^3)$

問題6

page 179, Section 2.4 Exercise 34

Compute each of these double sums.
  1. $\sum\limits_{i=1}^3 \sum\limits_{j=1}^2 (i-j)$ $= \sum\limits_{i=1}^3 [(i-1)+(i-2)]$$ = \sum\limits_{i=1}^3 (2i-3)$
  2. $\sum\limits_{i=0}^3 \sum\limits_{j=0}^2 (3i+2j)$ $= \sum\limits_{i=0}^3 [(3i+0)+(3i+2)+(3i+4)]$$ = \sum\limits_{i=0}^3 (9i+6)$
  3. $\sum\limits_{i=1}^3 \sum\limits_{j=0}^2 j$ $= \sum\limits_{i=1}^3 3$
  4. $\sum\limits_{i=1}^2 \sum\limits_{j=0}^3 (i^2j^3)$ $= \sum\limits_{i=1}^2 [0 + (i^2)+(8i^2)+(27i^2)]$$ = \sum\limits_{i=1}^2 (36i^2)$
  1. (1 − 1) + (1 − 2) + (2 − 1) + (2 − 2) + (3 − 1) + (3 − 2) = 3
  2. (0 + 0) + (0 + 2) + (0 + 4) + (3 + 0) + (3 + 2) + (3 + 4) + (6 + 0) + (6 + 2) +
    (6 + 4) + (9 + 0) + (9 + 2) + (9 + 4) = 78
  3. (0 + 1 + 2) + (0 + 1 + 2) + (0 + 1 + 2) = 9
  4. (0 + 1 + 8 + 27) + (0 + 4 + 32 + 108) = 180
Question 7

問題7

page 186, chapter 2.5 Exercises 2

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.
  1. the integers greater than 10
  2. the odd negative integers
  3. the integers with absolute value less than 1,000,000
  4. the real numbers between 0 and 2
  5. the set $A \times Z^+$ where $A = \{2, 3\}$
  6. the integers that are multiples of 10
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.
  1. the integers greater than 10
  2. the odd negative integers
  3. the integers with absolute value less than 1,000,000
  4. the real numbers between 0 and 2
  5. the set $A \times Z^+$ where $A = \{2, 3\}$
  6. the integers that are multiples of 10

  1. This set is countably infinite. The integers in the set are 11, 12, 13, 14, and so on. We can list these numbers in that order, thereby establishing the desired correspondence. In other words, the correspondence is given by $1\leftrightarrow11$, $2\leftrightarrow12$, $3\leftrightarrow13$, and so on; in general $n\leftrightarrow(n + 10)$.
  2. This set is countably infinite. The integers in the set are -1, -3, -5, -7, and so on. We can list these numbers in that order, thereby establishing the desired correspondence. In other words, the correspondence is given by $1\leftrightarrow-1$, $2\leftrightarrow-3$, $3\leftrightarrow-5$, and so on; in general $n\leftrightarrow-(2n - 1)$.
  3. This set is $\{-999,999, -999,998, \dots, -1, 0, 1, \dots, 999,999\}$. It is finite, with cardinality 1,999,999.
  4. This set is uncountable. We can prove it by the same diagonalization argument as was used to prove that the set of all reals is uncountable in Example 5.
  5. This set is countable. We can list its elements in the order $(2, 1), (3, 1), (2, 2), (3, 2), (2, 3), (3, 3), \dots$, giving us the one-to-one correspondence $1\leftrightarrow(2, 1)$, $2\leftrightarrow(3, 1)$, $3\leftrightarrow(2, 2)$, $4\leftrightarrow(3, 2)$, $5\leftrightarrow(2, 3)$, $6\leftrightarrow(3, 3), \dots$ .
  6. This set is countable. The integers in the set are 0, $\pm10, \pm20, \pm30$, and so on. We can list these numbers in the order 0, 10, -10, 20, -20, 30, $\dots$, thereby establishing the desired correspondence. In other words, the correspondence is given by $1\leftrightarrow0$, $2\leftrightarrow10$, $3\leftrightarrow-10$, $4\leftrightarrow20$, $5\leftrightarrow-20$, $6\leftrightarrow30$, and so on.
    7.

Question 8

問題8

page 186, Section 2.5 Exercise 6

Suppose that Hilbert's Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel

問題8

page 186, Section 2.5 Exercise 6

Suppose that Hilbert's Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel
Suppose that the hotel has infinite rooms and guests can't share room with other guests.

We want a one-to-one function from the set of positive integers to the set of odd positive integers. The simplest one to use is $f(n)=2n-1$ .We put the guest currently in Room $n$ into Room$(2n-1)$.Thus the guest in Room 1 stays put,the guest in Room 2 moves to Room 3,the guest in Room 3 moves to Room 5,and so on.

Question 9

問題9

page 194, chapter 2.6 Exercises 20

Let \[A = \left[\begin{matrix} -1 & 2 \\ 1 & 3 \\ \end{matrix}\right].\]
  1. Find $A^{-1}$. [Hint: Use Exercise 19.]
  2. Find $A^3$.
  3. Find $(A^{-1})^3$.
  4. Use your answers to (b) and (c) to show that $(A^{-1})^3$ is the inverse of $A^3$.
Let \[A = \left[\begin{matrix} -1 & 2 \\ 1 & 3 \\ \end{matrix}\right].\]
  1. Find $A^{-1}$. [Hint: Use Exercise 19.]
  2. Find $A^3$.
  3. Find $(A^{-1})^3$.
  4. Use your answers to (b) and (c) to show that $(A^{-1})^3$ is the inverse of $A^3$.

  1. Using Exercise 19, noting that $ad-bc = -5$, we write down the inverse immediately: \(\left[\begin{matrix} -\frac{3}{5} & \frac{2}{5} \\ \frac{1}{5} & \frac{1}{5} \\ \end{matrix}\right].\)
  2. We multiply to obtain \(A^2 = \left[\begin{matrix} 3 & 4 \\ 2 & 11 \\ \end{matrix}\right]\) and then \(A^3 = \left[\begin{matrix} 1 & 18 \\ 9 & 37 \\ \end{matrix}\right].\)
  3. We multiply to obtain \((A^{-1})^2 = \left[\begin{matrix} \frac{11}{25} & -\frac{4}{25} \\ -\frac{2}{25} & \frac{3}{25} \\ \end{matrix}\right]\) and then \((A^{-1})^3 = \left[\begin{matrix} -\frac{37}{125} & \frac{18}{125} \\ \frac{9}{125} & -\frac{1}{125} \\ \end{matrix}\right].\)
  4. Applying the method of Exercise 19 for obtaining inverses to the answer in part (b), we obtain the answer in part (c). Therefore $(A^3)^{-1} = (A^{-1})^3$.