discrete_mathematics_with_applications/chapter_7/exercises.md

Page 458

Exercise Set 7.1

1. Let X = \{1, 3, 5\} and Y = \{s, t, u, v\}. Define f: X \to Y by the following arrow diagram.

(See page 458 for image)

a. Write the domain of f and the co-domain of f.

Domain: \{1, 3, 5\}

Co-domain: \{s, t, u, v\}

b. Find f(1), f(3), and f(5).

f(1) = v, f(3) = s, f(5) = v

c. What is the range of f?

\{s, v\}

d. Is 3 an inverse image of s? Is 1 an inverse image of u?

yes; no

e. What is the inverse image of s? of u? of v?

\{3\};$\emptyset$;${1, 5}$

f. Represent f as a set of ordered pairs.

\{(1, v), (3, s), (5, v)\}

2. Let X = \{1, 3, 5\} and Y = \{a, b, c, d\}. Define g: X \to Y by the following arrow diagram.

(See page 459 for image)

a. Write the domain of g and the co-domain of g.

Domain: \{1, 3, 5\}

Co-domain: \{a, b, c, d\}

b. Find g(1), g(3), and g(5).

g(1) = b, g(3) = b, g(5) = b

c. What is the range of g?

\{b\}

d. Is 3 an inverse image of a? Is 1 an inverse image of b?

no;yes

e. What is the inverse image of b? of c?

\{1, 3, 5\}, \emptyset

f. Represent g as a set of ordered pairs.

 \{(1, b), (3, b), (5, b)\} 

3. Indicate whether the statements in parts (a)-(d) are true or false for all functions. Justify your answers.

a. If two elements in the domain of a function are equal, then their images in the co-domain are equal.

True. The definition of a function states that every input element in the domain must have an output element in the co-domain. Since two elements in the domain of the function are equal, then their outputs in the co-domain must be equal by this definition.

b. If two elements in the co-domain of a function are equal, then their preimages in the domain are also equal.

This is false. A function can have the same output for two different inputs.

c. A function can have the same output for more than one input.

True, the definition of a function only states that every input to the function must have an output, not necessarily unique outputs.

d. A function can have the same input for more than one output.

This is false. A single input can only map to a single output, not multiple outputs.

a. Find all functions from X = \{a, b\} to Y = \{u, v\}.

 f(a) = u, f(a) = v, f(b) = u, f(b) = v 

b. Find all functions from X = \{a, b, c\} to Y = \{u\}.

 f(a) = u, f(b) = u, f(c) = u 

c. Find all functions from X = \{a, b, c\} to Y = \{u, v\}.

 f(a) = u, f(a) = v, f(b) = u, f(b) = v, f(c) = u, f(c) k v 

5. Let I_{\mathbb{z}} bee the identity function defined on the set of all integers, and suppose that e, b_i^{jk}, K(t), and u_{kj} all represent integers. Find the following:

a. I_{\mathbb{Z}}(e)

 I_{\mathbb{Z}}(e) = e 

b. I_{\mathbb{Z}}\left(b_i^{jk}\right)

 I_{\mathbb{Z}}\left(b_i^{jk}\right) = b_i^{jk}\right 

c. I_{\mathbb{Z}}(K(t))

 I_{\mathbb{Z}}(K(t)) = K(t) 

d. I_{\mathbb{Z}}(u_{kj})

 I_{\mathbb{Z}}(u_{kj}) = u_{kj} 

6. Find functions defined on the set of nonnegative integers that can be used to define the sequences whose first six terms are given below.

a. 1, -\dfrac{1}{3}, \dfrac{1}{5}, -\dfrac{1}{7}, \dfrac{1}{9}, -\dfrac{1}{11}

 f: \mathbb{Z}^{\text{nonneg}} \to \mathbb{R} 

 f(n) = \frac{(-1)^n}{2n + 1} 

b. 0, -2, 4, -6, 8, -10

 f: \mathbb{Z}^{\text{nonneg}} \to \mathbb{R} 

 f(n) = (-1)^n \cdot 2n 

7. Let A = \{1, 2, 3, 4, 5\}, and define a function F: \mathscr{P}(A) \to \mathbb{Z} as follows: For each set X in \mathscr{P}(A),

F(x) = \begin{cases} 0& \text{if } X \text{ has an even number of elements} \ 1 & \text{if } X \text{ has an odd number of elements} \end{cases}

Find the following:

a. F(\{1, 3, 4\})

 F(\{1, 3, 4\}) = 1 

because \{1, 3, 4\} has an odd number of elements.

b. F(\emptyset)

 F(\emptyset) = 0 

because \emptyset has an even number of elements.

c. F(\{2, 3\})

 F(\{2, 3\}) = 0 

because \{2, 3\} has an even number of elements.

d. F(\{2, 3, 4, 5\})

 F(\{2, 3, 4, 5\}) = 0 

because \{2, 3, 4, 5\} has an even number of elements.

8. Let J_5 = \{0, 1, 2, 3, 4\}, and define a function F: J_5 \to J_5 as follows: For each x \in J_5, F(x) = (x^3 + 2x + 4) \mod 5.

Find the following:

a. F(0)

 F(0) = ((0)^3 + 2(0) + 4) \mod 5 

 = (0 + 0 + 4) \mod 5 

 = 4 \mod 5 

 = 4 

b. F(1)

 F(1) = ((1)^3 + 2(1) + 4) \mod 5 

 = (1 + 2 + 4) \mod 5 

 = 7 \mod 5 

 = 2 

c. F(2)

 F(2) = ((2)^3 + 2(2) + 4) \mod 5 

 = (8 + 4 + 4) \mod 5 

 = 16 \mod 5 

 = 1 

d. F(3)

 F(3) = ((3)^3 + 2(3) + 4) \mod 5 

 = (27 + 6 + 4) \mod 5 

 = 37 \mod 5 

 = 2 

e. F(4)

 F(4) = ((4)^3 + 2(4) + 4) \mod 5 

 = (64 + 8 + 4) \mod 5 

 = 76 \mod 5 

 = 1 

9. Define a function S: \mathbb{Z}^+ \to \mathbb{Z}^+ as follows: For each positive integer n,

 S(n) = \text{ the sum of the positive divisors of } n 

Find the following:

a. S(1)

 S(1) = 1 

b. S(15)

 S(15) = 1 + 3 + 5 + 15 = 24 

c. S(17)

 S(17) = 1 + 17 = 18 

d. S(5)

 S(5) = 1 + 5 = 6 

e. S(18)

 S(18) = 1 + 2 + 3 + 6 + 9 + 18  = 39 

f. S(21)

 S(21) = 1 + 3 + 7 + 21 = 32 

10. Let D be the set of all finite subsets of positive integers.

Define a function T: \mathbb{Z}^+ \to D as follows: For each positive integer n, T(n) = the set of positive divisors of n.

Find the following:

a. T(1)

 T(1) = \{1\} 

b. T(15)

 T(15) = \{1, 3, 5, 15\} 

c. T(17)

 T(17) = \{1, 17\} 

d. T(5)

 T(5) = \{1, 5\} 

e. T(18)

 T(18) = \{1, 2, 3, 6, 9, 18\} 

f. T(21)

 T(21) = \{1, 3, 7, 21\} 

11. Define F: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z} as follows: For every ordered pair (a, b) of integers, F(a, b) = (2a + 1, 3b - 2).

Find the following:

a. F(4, 4)

 F(4, 4) = (2(4) + 1, 3(4) - 2) 

 = (8 + 1, 12 - 2) 

 = (9, 10) 

b. F(2, 1)

 F(2, 1) = (2(2) + 1, 3(1) - 2) 

 = (4 + 1, 3 - 2) 

 = (5, 1) 

c. F(3, 2)

 F(3, 2) = (2(3) + 1, 3(2) - 2) 

 = (6 + 1, 6 - 2) 

 = (7, 4) 

d. F(1, 5)

 F(1, 5) = (2(1) + 1, 3(5) - 2) 

 = (2 + 1, 15 - 2) 

 = (3, 13) 

12. Let J_5 = \{0, 1, 2, 3, 4\}, and define G: J_5 \times J_5 \to J_5 \times J_5 as follows: For each (a, b) \in J_5 \times J_5,

 G(a, b) = ((2a + 1) \mod 5, (3b - 2) \mod 5) 

Find the following:

a. G(4, 4)

 G(4, 4) = ((2(4) + 1) \mod 5, (3(4) - 2) \mod 5) 

 = ((8 + 1) \mod 5, (12 - 2) \mod 5) 

 = (9 \mod 5, 10 \mod 5) 

 = (4, 0) 

b. G(2, 1)

 G(2, 1) = ((2(2) + 1) \mod 5, (3(1) - 2) \mod 5) 

 = ((4 + 1) \mod 5, (3 - 2) \mod 5) 

 = (5 \mod 5, 1 \mod 5) 

 = (0, 1) 

c. G(3, 2)

 G(3, 2) = ((2(3) + 1) \mod 5, (3(2) - 2) \mod 5) 

 = ((6 + 1) \mod 5, (6 - 2) \mod 5) 

 = (7 \mod 5, 4 \mod 5) 

 = (2, 4) 

d. G(1, 5)

 G(1, 5) = ((2(1) + 1) \mod 5, (3(5) - 2) \mod 5) 

 = ((2 + 1) \mod 5, (15 - 2) \mod 5) 

 = (3 \mod 5, 13 \mod 5) 

 = (3, 3) 

13. Let J_5 = \{0, 1, 2, 3, 4\}, and define functions f: J_5 \to J_5 and g: J_5 \to J_5 as follows: For each x \in J_5,

 f(x) = (x + 4)^2 \mod 5 \quad \text{ and } \quad g(x) = (x^2 + 3x + 1) \mod 5 

Is f = g? Explain.

x	f(x)	g(x)
0	1	1
1	0	0
2	1	1
3	4	4
4	4	4

The table shows that f(x) = g(x) for every x \in J_5. Therefore f = g by definition of equality of functions.

14. Define functions H and K from \mathbb{R} to \mathbb{R} by the following formulas:

For every x \in \mathbb{R},

 H(x) = \lfloor x \rfloor + 1 \quad \text{ and } \quad K(x) = \lceil x \rceil 

Does H = K? Explain.

No. For example say x = 0, then H(0) = \lfloor 0 \rfloor + 1 = 0 + 1 = 1 and K(0) = \lceil 0 \rceil = 0. Therefore it cannot be said that for every x \in \mathbb{R} that H(x) = K(x), and thus H \neq K.

15. Let F and G be functions from the set of all real numbers to itself. Define the product functions F \cdot G: \mathbb{R} \to \mathbb{R} and G \cdot F: \mathbb{R} \to \mathbb{R} as follows: For every x \in \mathbb{R},

 (F \cdot G)(x) = F(x) \cdot G(x) 

 (G \cdot F)(x) = G(x) \cdot F(x) 

Does F \cdot G = G \cdot F? Explain.

Yes, by the commutative law of multiplication of Real numbers:

 (F \cdot G)(x) = F(x) \cdot G(x) = G(x) \cdot F(x) = (G \cdot F)(x) 

Therefore, since (F \cdot G)(x) = (G \cdot F)(x) for all x \in \mathbb{R}, it can be concluded that F \cdot G = G \cdot F by the definition of equality of functions.

16. Let F and G be function sfrom the set of all real numbers to itself. Define new functions F - G: \mathbb{R} \to \mathbb{R} and G - F: \mathbb{R} \to \mathbb{R} as follows: For every x \in \mathbb{R},

 (F - G)(x) = F(x) - G(x) 

 (G - F)(x) = G(x) - F(x) 

Does F - G = G - F? Explain.

No. Consider the definition of the difference of sets:

 (F - G)(x) = F(x) - G(x) = F(x) 

and:

 (G - F)(x) = G(x) - F(x) = G(x) 

Since F(x) \neq G(x) for all x \in \mathbb{R}, it can be concluded that F - G \neq G - F by the definition of the equality of functions.

17. Use the definition of logarithm to fill in the blanks below.

a. \log_28 = 3 because _____.

 2^3 = 8 

b. \log_5\left(\dfrac{1}{25}\right) = -2 because _____.

 5^{-2} = \frac{1}{5^2} = \frac{1}{25} 

c. \log_44 = 1 because _____.

 4^1 = 4 

d. \log_3(3^n) = n because _____.

 3^n = 3^n 

e. \log_41 = 0 because _____.

 4^0 = 1 

18. Find exact values for each of the following quantities without using a calculator.

a. \log_{3}81

 3^{\text{?}} = 81 

 \log_{3}81 = 4 

b. \log_{2}1024

 2^{\text{?}} = 1024 

 \log_{2}1024 = 10 

c. \log_{3}\left(\dfrac{1}{27}\right)

 \log_{3}\left(\frac{1}{27}\right) = -3 

d. \log_{2}1

 \log_{2}1 = 0 

e. \log_{10}\left(\dfrac{1}{10}\right)

 \log_{10}\left(\dfrac{1}{10}\right) = -1 

f. \log_{3}3

 \log_{3}3 = 1 

g. \log_{2}(2^k)

\log_{2}(2^k) = k 

19. Use the definition of logarithm to prove that for any positive real number b with b \neq 1, \log_{b}b = 1.

Proof:

Let b be any positive real number with b \neq 1. Since b^1 = b, then \log_{b}b = 1 by definition of logarithm.

Q.E.D.

20. Use the definition of logarithm to prove that for any positive real number b with b \neq 1, \log_{b}1 = 0.

Proof:

Let b be any positive real number with b \neq 1. Since b^0 = 1, then \log_{b}1 = 0 by definition of logarithm.

Q.E.D.

21. If b is any positive real number with b \neq 1 and x is any real number, b^{-x} is defined as follows:

b^{-x} = \dfrac{1}{b^x}. Use this definition and the definition of logarithm to prove that \log_{b}\left(\dfrac{1}{u}\right) = -\log_{b}u for all positive real numbers u and b, with b \neq 1.

Proof:

Let b be any positive real number with b \neq 1. Let u be any positive real number.

Let v = \log_{b}\left(\dfrac{1}{u}\right). By the definition of logarithm, this means that b^v = \dfrac{1}{u}. It follows by algebra that:

 b^v = \frac{1}{u} 

 u \cdot b^v = 1 

 u = \frac{1}{b^v} 

 u = b^{-v} 

Hence, by the definition of logarithm:

 -v = \log_{b}(u) 

and by algebra:

 v = -\log_{b}(u) 

Since v = \log_{b}\left(\dfrac{1}{u}\right) and v = -\log_{b}(u), it follows by the definition of equality that:

 \log_{b}\left(\frac{1}{u}\right) = -\log{b}(u) 

This is what was to be shown.

Q.E.D.

22. Use the unique factorization for the integers theorem (Section 4.4) and the definition of logarithm to prove that \log_{3}(7) is irrational.

Hint: Use a proof by contradiction. Suppose \log_{3}7 is rational. Then \log_{3}7 = \dfrac{a}{b} for some integers a and b with b \neq 0.

Apply the definition of logarithm and rewrite \log_{3}7 = \dfrac{a}{b} in exponential form.

Proof (by contradiction):

Suppose \log_{3}(7) is rational, that is \log_{3}(7) = \dfrac{a}{b} for some integers a and b where b \neq 0.

By the definition of logarithm, this would mean that:

 3^{\frac{a}{b}} = 7 

Then by algebra:

 3^a = 7^b 

Since b \neq 0, we know that 7^b \neq 1, and by equality it follows that 3^a \neq 1. Additionally, by the definition of exponentiation, it is known that 7^b > 0 and 3^a > 0 (they are both positive numbers).

But, by the unique factorization for integers theorem, this means that 7^b and 3^a are two different prime factorizations of the same positive integer. This is only possible if the positive integer is equal to 1.

Hence 3^a = 7^b = 1, but it has already been established that 3^a = 7^b \neq 1. This is a contradiction.

Therefore the supposition is false, and \log_{3}(7) is irrational.

Q.E.D.

23. If b and y are positive real numbers such that \log_{b}y = 3, what is \log_{\frac{1}{b}}y? Explain.

Proof:

Suppose b and y are positive real numbers such that \log_{b}y = 3.

By the definition of logarithm, this means that:

 b^3 = y 

To find \log_{\frac{1}{b}}y, first, replace y by substitution:

 \log_{\frac{1}{b}}y 

 = \log_{\frac{1}{b}}(b^3) 

Then notice that \dfrac{1}{b} = b^{-1}, and then substitute:

 = \log_{b^{-1}}(b^3) 

By the definition of logarithm, this means that:

 (b^{-1})^x = b^3 

Where x is \log_{\frac{1}{b}}y, or our answer. By the multiplication of exponents, this means that:

 b^{-1 \cdot x} = b^3 

And by multiplication of negative numbers:

 b^{-1 \cdot -3} = b^3 

Therefore x = -3, or:

 \log_{\frac{1}{b}}y = -3 

This is what was to be found.

Q.E.D.

24. If b and y are positive real numbers such that \log_{b}y = 2, what is \log_{b^2}(y)? Explain.

Proof:

Suppose b and y are positive real numbers such that \log_{b}y = 2. By the definition of logarithm, this means that:

 \log_{b}y = 2 

 b^2 = y 

To find \log_{b^2}(y), first substitute in for y:

 \log_{b^2}(b^2) 

By the definition of logarithm, this means that:

 \log_{b^2}(b^2) = 1 

because (b^2)^1 = b^2.

This is what was to be found.

Q.E.D.

25. Let A = \{2, 3, 5\} and B = \{x, y\}. Let p_1 and p_2 be the projections of A \times B onto the first and second coordinates. That is, for each pair (a, b) \in A \times B, p_1(a, b) = a and p_2(a, b) = b.

a. Find p_1(2, y) and p_1(5, x). What is the range of p_1?

 p_1(2, y) = 2 

 p_1(5, x) = 5 

Range of p_1:

 \{2, 3, 5\} 

b. Find p_2(2, y) and p_2(5, x). What is the range of p_2?

 p_2(2, y) = y 

 p_2(5, x) = x 

Range of p_2:

 \{x, y\} 

26. Observe that \mod and \text{div} can be defined as functions from \mathbb{Z}^{\text{nonneg}} \times \mathbb{Z}^+$ to \mathbb{Z}. For each ordered pair (n, d) consisting of a nonnegative integer n and a positive integer d, let

\mod(n, d) = n \mod d (the nonnegative remainder obtained when n is divided by d).

\text{div}(n, d) = n \text{ div } d (the integer quotient obtained when n is divided by d).

Find each of the following:

a. \mod(67, 10) and \text{div}(67, 10)

 \mod(67, 10) = 7 

 \text{div}(67, 10) = 6 

b. \mod(59, 8) and \text{div}(59, 8)

 \mod(59, 8) = 3 

 \text{div}(59, 8) = 7 

c. \mod(30, 5) and \text{div}(30, 5)

 \mod(30, 5) = 0 

 \text{div}(30, 5) = 6 

27. Let S be the set of all strings of $a$'s and $b$'s.

a. Define f: S \to \mathbb{Z} as follows: For each string s in S

f(s) = \begin{cases} & \text{ the number of b's to the left-most a in s} \ 0 & \text{if s contains no a's} \end{cases}

Find f(aba), f(bbab), and f(b). What is the range of f?

 f(aba) = 0 

 f(bbab) = 2 

 f(b) = 0 

The range of f: \mathbb{Z}^{\text{nonneg}}

b. Define g: S \to S as follows: For each string s in S,

 g(s) = \text{ the string obtained by writing the characters of s in reverse order} 

Find g(aba), g(bbab), and g(b). What is the range of g?

 g(aba) = aba 

 g(bbab) = babb 

The range of g is S.

28. Consider the coding and decoding functions E and D defined in Example 7.1.9.

a. Find E(0110) and D(111111000111).

 E(0110) = 000111111000 

 D(111111000111) = 1101 

b. Find E(1010) and D(000000111111).

 E(1010) = 111000111000 

 D(000000111111) = 0011 

29. Consider the Hamming distance function defined in Example 7.1.10.

a. Find H(10101, 00011).

 H(10101, 00011) = 3 

b. Find H(00110, 10111).

 H(00110, 10111) = 2 

30. Draw arrow diagrams for the Boolean functions defined by the following input/output tables.

a.

Input	Intput	Output
P	Q	R
-------	-
1	1	0
1	0	1
0	1	0
0	0	1

Omitted.

b.

Input	Intput	Input	Output
P	Q	R	S
-	-	-	-
1	1	1	1
1	1	0	0
1	0	1	1
1	0	0	1
0	1	1	0
0	1	0	0
0	0	1	0
0	0	0	1

Omitted.

31. Fill in the following table to show the values of all possible two-place Boolean functions.

Input	Input	f_1	f_2	f_3	f_4	f_5	f_6	f_7	f_8	f_9	f_{10}	f_{11}	f_{12}	f_{13}	f_{14}	f_{15}	f_{16}
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
1	0	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
0	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1

32. Consider the three-place Boolean function f defined by the following rule: For each triple (x_1, x_2, x_3) of $0$'s and $1$'s,

 f(x_1, x_2, x_3) = (4x_1 + 3x_2 + 2x_3) \mod 2 

a. Find f(1, 1, 1) and f(0, 0, 1).

 f(1, 1, 1) = (4(1) + 3(1) + 2(1)) \mod 2 

 f(1, 1, 1) = (4 + 3 + 2) \mod 2 

 f(1, 1, 1) = 9 \mod 2 

 f(1, 1, 1) = 1 

 f(0, 0, 1) = (4(0) + 3(0) + 2(1)) \mod 2 

 f(0, 0, 1) = (0 + 0 + 2) \mod 2 

 f(0, 0, 1) = 2 \mod 2 

 f(0, 0, 1) = 0 

b. Describe f using an input/output table.

x_1	x_2	x_3	f(x_1, x_2, x_3)
0	0	0	0
0	0	1	0
0	1	0	1
0	1	1	1
1	0	0	0
1	0	1	0
1	1	0	1
1	1	1	1

33. Student A tries to define a function g: \mathbb{Q} \to \mathbb{Z} by the rule

g\left(\dfrac{m}{n}\right) = m - n, for all integers m and n with n \neq 0.

Student B claims that g is not well defined. Justify student B's claim.

Suppose \dfrac{m}{n} = \dfrac{1}{2}, this would mean that g\left(\dfrac{m}{n}\right) = 1 - 2 = -1.

Since \dfrac{m}{n} = \dfrac{1}{2}, this means that \dfrac{m}{n} = \dfrac{1}{2} = \dfrac{2}{4}. Since they are equivalent, this means that g\left(\dfrac{1}{2}\right) = g\left(\dfrac{2}{4}\right) = 2 - 4 = -2.

But notice that:

 g\left(\frac{1}{2}\right) = -1 \neq -2 = g\left(\frac{2}{4}\right) 

Since the function g gives two different outputs for the same input, the function g is not well defined.

34. Student C tries to define a function h: \mathbb{Q} \to \mathbb{Q} by the rule

h\left(\dfrac{m}{n}\right) = \dfrac{m^2}{n}, for all integers m and n with n \neq 0.

Student D claims that h is not well defined. Justify student D's claim.

Suppose \dfrac{m}{n} = \dfrac{2}{3}, then h\left(\dfrac{2}{3}\right) = \dfrac{(2)^2}{3} = \dfrac{4}{3}.

Notice that \dfrac{2}{3} = \dfrac{4}{6}, so h\left(\dfrac{4}{6}\right) = \dfrac{(4)^2}{6} = \dfrac{16}{6} = \dfrac{8}{3}.

Notice that:

 h\left(\frac{2}{3}\right) = \frac{4}{3} \neq \frac{8}{3} = h\left(\frac{4}{6}\right) 

Since the function h does not produce the same output given the same input, the function is not well defined.

35. Let U = \{1, 2, 3, 4\}. Student A tries to define a function R: U \to \mathbb{Z} as follows: For each x \in U,

R(x) is the integer y so that (xy) \mod 5 = 1.

Student B claims that R is not well defined. Who is correct: student A or student B? Justify your answer.

Consider R(3) = 2 since (3 \cdot 2) \mod 5 = 1. On the other hand, R(3) = 7 since (3 \cdot 7) \mod 5 = 1.

Since R returns multiple outputs for the same input, it is not well defined, and Student B is correct.

36. Let V = \{1, 2, 3\}. Student C tries to define a function S: V \to V as follows: For each x \in V,

S(x) is the integer y in V so that (xy) \mod 4 = 1.

Student D claims that S is not well defined. Who is right: student C or student D? Justify your answer.

Consider S(1) = 17 since (1 \cdot 17) \mod 4 = 1. On the other hand S(1) = 13 since (1 \cdot 13) \mod 4 = 1.

Since S returns multiple outputs for the same input, it is not well defined, and Student D is correct.

37. On certain computers the integer data type goes from -2,147,483,648 through 2,147,483,647. Let S be the set of all integers from -2,147,483,648 through 2,147,483,647. Try to define a function f: S \to S by the rule f(n) = n^2 for each n in S. Is f well defined? Explain.

No, 2,147,483,247 = 2^{31} - 1, so for values of n greater than 2^{16}, f(n) = n^2 will be greater than 2^{32}, which falls outside of S.

38. Let X = \{a, b, c\} and Y = \{r, s, t, u, v, w\}. Define f: X \to Y as follows: f(a) = v, f(b) = v, and f(c) = t.

a. Draw an arrow diagram for f.

Omitted.

b. Let A = \{a, b\}, C = \{t\}, D = \{u, v\}, and E = \{r, s\}. Find f(A), f(X), f^{-1}(C), f^{-1}(D), f^{-1}(E), and f^{-1}(Y).

 f(A) = \{v\} 

 f(X) = $\{t, v\} 

 f^{-1}(C) = \{c\} 

 f^{-1}(D) = \{a, b\} 

 f^{-1}(E) = \emptyset 

 f^{-1}(Y) = \{a, b, c\} 

39. Let X = \{1, 2, 3, 4\} and Y = \{a, b, c, d, e\}. Define g: X \to Y as follows: g(1) = a, g(2) = a, g(3) = a, and g(4) = d.

a. Draw an arrow diagram for g.

Omitted.

b. Let A = \{2, 3\}, C = \{a\}, and D = \{b, c\}. Find g(A), g(X), g^{-1}(C), g^{-1}(D), and g^{-1}(Y).

 g(A) = \{a\} 

 g(X) = \{a, d\} 

 g^{-1}(C) = \{1, 2, 3\} 

 g^{-1}(D) = \emptyset 

 g^{-1}(Y) = \{1, 2, 3, 4\} 

40. Let X and Y be sets, let A and B be any subsets of X, and let F be a function from X to Y. Fill in the blanks in the following proof that F(A) \cup F(B) \subseteq F(A \cup B).

Proof:

Let y be any element in F(A) \cup F(B). [We must show that y is in F(A \cup B).] By definition of union, __ (i) __.

Case 1 y \in F(A):

In this case, by definition of F(A), y = F(x) for __ (ii) __ x \in A. Since A \subseteq A \cup B, it follows from the definition of union that x \in __ (iii) __. Hence, y = F(x) for some x \in A \cup B, and thus, by definition of F(A \cup B), y \in __ (iv) __.

Case 2, y \in F(B):

In this case, by definition of F(B), __ (v) __ for some x \in B. Since B \subseteq A \cup B it follows from the definition of union that __ (vi) __. Thus y \in F(A \cup B).

Therefore, regardless of whether y \in F(A) or y \in F(B), we have that y \in F(A \cup B) [as was to be shown].

i. y \in F(A) \cup F(B)

ii. some

iii. A \cup B

iv. F(A \cup B)

v. y = F(x)

vi. x \in A \cup B

In 41-49 let X and Y be sets, let A and B be any subsets of X, and let C and D be any subsets of Y. Determine which of the properties are true for every function F from X to Y and which are false for at least one function F from X to Y. Justify your answers.

41. If A \subseteq B then F(A) \subseteq F(B)

Proof:

Let F be a function from X to Y and suppose A \subseteq X, B \subseteq X, and A \subseteq B.

Then, let y be some element such that y \in F(A).

By definition of image of a set, y = F(x) for some x \in A. Thus since A \subseteq B, x \in B, and so y = F(x) for some x \in B. Hence y \in F(B), and therefore F(A) \subseteq F(B).

Q.E.D.

42. F(A \cap B) \subseteq F(A) \cap F(B)

Proof:

Suppose y is some element such that y \in F(A \cap B).

By the supposition and the definition of A \cap B, this means that y = F(x) for some x \in A \cap B.

By the definition of intersection, it follows that x \in A and x \in B.

By the definition of F(A) and F(B), y = F(x) is in F(A) and in F(B).

Hence, by the definition of intersection, y \in F(A) \cap F(B).

Since y \in F(A) \cap F(B), it can be concluded that F(A \cap B) \subseteq F(A) \cap F(B).

Q.E.D.

43. F(A) \cap F(B) \subseteq F(A \cap B)

Disproof (by counterexample):

Let X = \{1, 2, 3\} and Y = \{a, b\}. Then, define a function F: X \to Y such that F(1) = a, F(2) = b, F(3) = b.

Let A = \{1, 2\} and B = \{1, 3\}. Then F(A) = \{a, b\} and F(B) = \{a, b\}.

So F(A) \cap F(B) = \{a, b\}, and F(A \cap B) = F(\{1\}) = \{a\}.

Since \{a\} \neq \{a, b\}, the given statement is false.

Q.E.D.

44. For all subsets A and B of X, F(A - B) = F(A) - F(B).

Disproof (by counterexample):

Let X = \{1, 2\} and Y = \{a\}. Then, define a function F: X \to Y such that F(1) = a and F(2) = a.

Let A = \{1\} and B = \{2\}. Then F(A - B) = F(\{1\}) = \{a\}.

Then F(A) - F(B) = F(\{1\}) - F(\{2\}) = \{a\} - \{a\} = \emptyset.

Since \{a\} \neq \emptyset, the given statement is false.

Q.E.D.

45. For all subsets C and D of Y, if C \subseteq D, then F^{-1}(C) \subseteq F^{-1}(D).

Proof:

Let F be a function from a set X to a set Y, and suppose C \subseteq Y, D \subseteq Y, and C \subseteq D.

Suppose x \in F^{-1}(C). Then F(x) \in C. Since C \subseteq D, F(x) \in D also. Hence, by definition of inverse image, x \in F^{-1}(D). Therefore F^{-1}(C) \subseteq F^{-1}(D).

Q.E.D.

46. For all subsets C and D of Y,

 F^{-1}(C \cup D) = F^{-1}(C) \cup F^{-1}(D) 

Proof:

In order to prove:

 F^{-1}(C \cup D) = F^{-1}(C) \cup F^{-1}(D) 

We must prove:

 F^{-1}(C \cup D) \subseteq F^{-1}(C) \cup F^{-1}(D) 

and:

 F^{-1}(C) \cup F^{-1}(D) \subseteq F^{-1}(C \cup D) 

Proof F^{-1}(C \cup D) \subseteq F^{-1}(C) \cup F^{-1}(D):

Suppose x \in F^{-1}(C \cup D). Then F(x) \in C \cup D. By the definition of union, this means that F(x) \in C or F(x) \in D.

Case F(x) \in C:

Since F(x) \in C, this means that x \in F^{-1}(C). By the definition of union, this means that x \in F^{-1}(C) \cup F^{-1}(D).

Case F(x) \in D:

Since F(x) \in D, this means that x \in F^{-1}(D). By the definition of union, this means that x \in F^{-1}(C) \cup F^{-1}(D).

In both cases, x \in F^{-1}(C) \cup F^{-1}(D). Therefore, any element in F^{-1}(C \cup D) is also in F^{-1}(C), and F^{-1}(C \cup D) \subseteq F^{-1}(C) \cup F^{-1}(D), as was to be shown.

Proof F^{-1}(C) \cup F^{-1}(D) \subseteq F^{-1}(C \cup D):

Suppose x \in F^{-1}(C) \cup F^{-1}(D). By definition of union, this means that x \in F^{-1}(C) or x \in F^{-1}(D).

Case x \in F^{-1}(C):

Since x \in F^{-1}(C), this means that F(x) \in C. It follows by definition of union that F(x) \in C \cup D, or x \in F^{-1}(C \cup D).

Case x \in F^{-1}(D):

Since x \in F^{-1}(D), this means that F(x) \in D. It follows by definition of union that F(x) \in C \cup D, or x \in F^{-1}(C \cup D).

In both cases, x \in F^{-1}(C \cup D). Therefore any element in F^{-1}(C) \cup F^{-1}(D) is in F^{-1}(C \cup D), and so F^{-1}(C) \cup F^{-1}(D) \subseteq F^{-1}(C \cup D). This is what was to be shown.

Conclusion:

Since both subset relations have been proved, it can be concluded that F^{-1}(C \cup D) = F^{-1}(C) \cup F^{-1}(D). This is what was to be shown.

Q.E.D.

47. For all subsets C and D of Y,

 F^{-1}(C \cap D) = F^{-1}(C) \cap F^{-1}(D) 

Proof:

In order to prove:

 F^{-1}(C \cap D) = F^{-1}(C) \cap F^{-1}(D) 

it must be shown that:

 F^{-1}(C \cap D) \subseteq F^{-1}(C) \cap F^{-1}(D) 

and also that:

 F^{-1}(C) \cap F^{-1}(D) \subseteq F^{-1}(C \cap D) 

Proof F^{-1}(C \cap D) \subseteq F^{-1}(C) \cap F^{-1}(D):

Suppose x \in F^{-1}(C \cap D), or F(x) \in C \cap D. By definition of intersection, this means that F(x) \in C and F(x) \in D, or x \in F^{-1}(C) \cap F^{-1}(D). This is what was to be shown.

Proof F^{-1}(C) \cap F^{-1}(D) \subseteq F^{-1}(C \cap D):

Suppose x \in F^{-1}(C) \cap F^{-1}(D), or F(x) \in C and F(x) \in D. By definition of intersection, F(x) \in C \cap D, or x \in F^{-1}(C \cap D). This is what was to be shown.

Conclusion:

Since both subset relations have been proved, it can be concluded that F^{-1}(C \cap D) = F^{-1}(C) \cap F^{-1}(D). This is what was to be shown.

Q.E.D.

48. For all subsets C and D of Y,

 F^{-1}(C - D) = F^{-1}(C) - F^{-1}(D) 

Proof:

In order to prove:

 F^{-1}(C - D) = F^{-1}(C) - F^{-1}(D) 

it must be shown that:

 F^{-1}(C - D) \subseteq F^{-1}(C) - F^{-1}(D) 

and also that:

 F^{-1}(C) - F^{-1}(D) \subseteq F^{-1}(C - D) 

Proof F^{-1}(C - D) \subseteq F^{-1}(C) - F^{-1}(D):

Suppose x \in F^{-1}(C - D), or F(x) \in C - D. By definition of difference of sets, this means that F(x) \in C and F(x) \notin D. By the definition of inverse image, this means x \in F^{-1}(C) and x \notin F^{-1}(D). By the definition of difference, this is x \in F^{-1}(C) - F^{-1}(D). Thus F^{-1}(C - D) \subseteq F^{-1}(C) - F^{-1}(D), which is what was to be shown.

Proof F^{-1}(C) - F^{-1}(D) \subseteq F^{-1}(C - D):

Suppose x \in F^{-1}(C) - F^{-1}(D), or F(x) \in C and F(x) \notin D. By the definition of inverse image, this means that F(x) \in C - D, or x \in F^{-1}(C - D). Thus F^{-1}(C) - F^{-1}(D) \subseteq F^{-1}(C - D), which is what was to be shown.

Conclusion:

Since both subset relations have been proved, it can be concluded that F^{-1}(C - D) = F^{-1}(C) - F^{-1}(D), which is what was to be shown.

Q.E.D.

49. F(F^{-1}(C)) \subseteq C

Proof:

Suppose x \in F(F^{-1}(C)). By definition of image, there exists some a \in F^{-1}(C) such that F(a) = x. By definition of inverse image, a \in F^{-1}(C) means F(a) \in C. Since F(a) = x, we have x \in C. Therefore F(F^{-1}(C)) \subseteq C.

Q.E.D.

50. Given a set S and a subset A, the characteristic function of $A$, denoted \chi_A, is the function defined from S to \mathbb{Z} with the property that for each u \in S,

\chi_{A}(u) = \begin{cases} 1 & \text{if } u \in A \ 0 & \text{if } u \notin A \end{cases}

Show that each of the following holds for all subsets A and B of S and every u \in S.

a. \chi_{A \cap B}(u) = \chi_{A}(u) \cdot \chi_{B}(u)

Omitted.

b. \chi_{A \cup B}(u) = \chi_{A}(u) + \chi_{B}(u) - \chi_{A}(u) \cdot \chi_{B}(u)

Omitted.

Each of exercises 51-53 refers to the Euler phi function, denoted \phi, which is defined as follows: For each integer n \geq 1, \phi(n) is the number of positive integers less than or equal to n that have no common factors with n except \pm 1. For example \phi(10) = 4 because there are four positive integers less than or equal to 10 that have no common factors with 10 except \pm 1 - namely, 1, 3, 7, and 9.

51. Find each of the following:

a. \phi(15)

Omitted.

b. \phi(2)

Omitted.

c. \phi(5)

Omitted.

d. \phi(12)

Omitted.

e. \phi(11)

Omitted.

f. \phi(1)

Omitted.

52. Prove that if p is a prime number and n is an integer with n \geq 1, then \phi(p^n) = p^n - p^{n - 1}.

Omitted.

53. Prove that there are infinitely many integers n for which \phi(n) is a perfect square.

Omitted.

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Page 480

Exercise Set 7.2

1. The definition of one-to-one is stated in two ways:

 \forall x_1, x_2 \in X, \text{ if } F(x_1) = F(x_2) \tex{ then } x_1 = x_2 

and

 \forall x_1, x_2 \in X, \text{ if } x_1 \neq x_2 \tex{ then } F(x_1) \neq F(x_2) 

Why are these two statements logically equivalent?

2. Fill in each blank with the word most or least.

a. A function F is one-to-one if, and only if, each element in the co-domain of F is the image of at _____ one element in the domain of F.

b. A function F is onto if, and only if, each element in the co-domain of F is the image of at _____ one element in the domain of F.

3. When asked to state the definition of one-to-one, a student replies, "A function f is one-to-one if, and only if, every element of X is sent by f to exactly one element of Y." Give a counterexample to show that the student's reply is incorrect.

4. Let f: X \to Y be a function. True or false? A sufficient condition for f to be one-to-one is that for every element y in Y, there is at most one x in X with f(x) = y. Explain your answer.

5. All but two of the following statements are correct ways to express the fact that a function f is onto. Find the two that are incorrect.

a. f is onto \Leftrightarrow every element in its co-domain is the image of some element in its domain.

b. f is onto \Leftrightarrow every element in its domain has a corresponding image in its co-domain.

c. f is onto \Leftrightarrow \forall y \in Y, \exists x \in X such that f(x) = y.

d. f is onto \Leftrightarrow \forall x \in X, \exists y \in Y such that f(x) = y.

e. f is onto \Leftrightarrow the range of f is the same as the co-domain of f.

6. Let X = \{1, 5, 9\} and Y = \{3, 4, 7\}.

a. Define f: X \to Y by specifying that

 f(1) = 4, f(5) = 7, f(9) = 4 

Is f one-to-one? Is f onto? Explain your answers.

b. Define g: X \to Y by specifying that

 g(1) = 7, g(5) = 3, g(9) = 4 

Is g one-to-one? Is g onto? Explain your answers.

7. Let X = \{a, b, c, d\} and Y = \{e, f, g\}. Define functions F and G by the arrow diagrams below.

(See page 481) for images.

a. Is F one-to-one? Why or why not? Is it onto? Why or why not?

b. Is G one-to-one? Why or why not? Is it onto? Why or why not?

8. Let X = \{a, b, c\} and Y = \{d, e, f, g\}. Define functions H and K by the arrow diagrams below.

(See page 481) for images.

a. Is H one-to-one? Why or why not? Is it onto? Why or why not?

b. Is K one-to-one? Why or why not? Is it onto? Why or why not?

9. Let X = \{1, 2, 3\}, Y = \{1, 2, 3, 4\}, and Z = \{1, 2\}.

a. Define a function f: X \to Y that is one-to-one but not onto.

b. Define a function g: X \to Z that is onto but not one-to-one.

c. Define a function h: X \to X that is neither one-to-one nor onto.

d. Define a function k: X \to X that is one-to-one and onto but is not the identity function on X.

a. Define f: \mathbb{Z} \to \mathbb{Z} by the rule f(n) = 2n, for every integer n.

i. Is $f$ one-to-one? Prove or give a counterexample.

ii. Is $f$ onto? prove or give a counterexample.

b. Let 2\mathbb{Z} denote the set of all even integers. That is, 2\mathbb{Z} = \{n \in \mathbb{Z} | n = 2k \text{, for some integer } k\}. Define h: \mathbb{Z} \to 2\mathbb{Z} by the rule h(n) = 2n, for each integer n. Is h onto? Prove or give a counterexample.

a. Define g: \mathbb{Z} \to \mathbb{Z} by the rule g(n) = 4n - 5, for each integer n.

i. Is $g$ one-to-one? Prove or give a counterexample.

ii. Is $g$ onto? Prove or give a counterexample.

b. Define G: \mathbb{R} \to \mathbb{R} by the rule G(x) = 4x - 5 for every real number x. Is G onto? Prove or give a counterexample.

a. Define F: \mathbb{Z} \to \mathbb{Z} by the rule F(n) = 2 - 3n, for each integer n.

i. Is $F$ one-to-one? Prove or give a counterexample.

ii. Is $F$ onto? Prove or give a counterexample.

b. Define G: \mathbb{R} \to \mathbb{R} by the rule G(x) = 2 - 3x for each real number x. Is G onto? Prove or give a counterexample.

a. Define H: \mathbb{R} \to \mathbb{R} by the rule H(x) = x^2, for each real number x.

i. Is $H$ one-to-one? Prove or give a counterexample.

ii. Is $H$ onto? Prove or give a counterexample.

b. Define K: \mathbb{R}^{\text{nonneg}} \to \mathbb{R}^{\text{nonneg}} by the rule K(x) = x^2, for each nonnegative real number x. Is K onto? Prove or give a counterexample.

14. Explain the mistake in the following "proof."

Theorem: The function f: \mathbb{Z} \to \mathbb{Z} defined by the formula f(n) = 4n + 3, for each integer n, is one-to-one.

"Proof: Suppose any integer n is given. Then by definition of f, there is only one possible value for f(n) - namely, 4n + 3. Hence f is one-to-one."

In each of 15-18 a function f is defined on a set of real numbers. Determine whether or not f is one-to-one and justify your answer.

15. f(x) = \dfrac{x + 1}{x}, for each number x \neq 0

16. f(x) = \dfrac{x}{x^2 + 1}, for each real number x

17. f(x) = \dfrac{3x - 1}{x}, for each real number x \neq 0

18. f(x) = \dfrac{x + 1}{x - 1}, for each real number x \neq 1

19. Referring to Example 7.2.3, assume that records with the following ID numbers are to be placed in sequence into Table 7.2.1. Find the position into which each record is placed.

a. 417302072

b. 364981703

c. 283090787

20. Define \text{Floor}: \mathbb{R} \to \mathbb{Z} by the formula \text{Floor}(x) = \lfloor x \rfloor, for every real number x.

a. Is \text{Floor} one-to-one? Prove or give a counterexample.

b. Is \text{Floor} onto? Prove or give a counterexample.

21. Let S be the set of all strings of $0$'s and $1$'s, and define L: S \to \mathbb{Z}^{\text{nonneg}} by

 L(s) = \text{ the length of } s \text{, for every string } s \text{ in } S 

a. Is L one-to-one? Prove or give a counterexample.

b. Is L onto? Prove or give a counterexample.

22. Let S be the set of all strings of $0$'s and $1$'s, and define D: S \to \mathbb{Z} as follows: For every s \in S,

 D(s) = \text{ the number of 1's in } s \text{ minus the number of 0's in } s 

a. Is D one-to-one? Prove or give a counterexample.

b. Is D onto? Prove or give a counterexample.

23. Define F: \mathscr{P}(\{a, b, c\}) \to \mathbb{Z} as follows: For every A in \mathscr{P}(\{a, b, c\}),

 F(A) = \text{ the number of elements in } A 

a. Is F one-to-one? Prove or give a counterexample.

b. Is F onto? Prove or give a counterexample.

24. Let S be the set of all strings of $a$'s and $b$'s, and define N: S \to \mathbb{Z} by

 N(s) = \text{ the number of a's in } s \text{, for each } s \in S 

a. Is N one-to-one? Prove or give a counterexample.

b. Is N onto? Prove or give a counterexample.

25. Let S be the set of all strings in $a$'s and $b$'s, and define C: S \to S by

 C(s) = as \text{, for each } s \in S 

(C is called concatenation by a on the left.)

a. Is C one-to-one? Prove or give a counterexample.

b. Is C onto? Prove or give a counterexample.

26. Define S: \mathbb{Z}^+ \to \mathbb{Z}^+ by the rule: For each integer n,

 S(n) = \text{ the sum of the positive divisors of } n 

a. Is S one-to-one? Prove or give a counterexample.

b. Is S onto? Prove or give a counterexample.

27. Let D be the set of all finite subsets of positive integers, and define T: \mathbb{Z}^+ \to D by the following rule:

For every integer n, T(n) = \text{ the set of all of the positive divisors of } n.

a. Is T one-to-one? Prove or give a counterexample.

b. Is T onto? Prove or give a counterexample.

28. Define G: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R} as follows:

 G(x, y) = (2y, -x) \text{ for every } (x, y) \in \mathbb{R} \times \mathbb{R} 

a. Is G one-to-one? Prove or give a counterexample.

b. Is G onto? Prove or give a counterexample.

29. Define H: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R} as follows:

 H(x, y) = (x + 1, 2 - y) \text{ for every } (x, y) \in \mathbb{R} \times \mathbb{R} 

a. Is H one-to-one? Prove or give a counterexample.

b. Is H onto? Prove or give a counterexample.

30. Define J: \mathbb{Q} \times \mathbb{Q} \to \mathbb{R} by the rule

 J(r, s) = r + \sqrt{2}s \text{ for each } (r, s) \in \mathbb{Q} \times \mathbb{Q} 

a. Is J one-to-one? Prove or give a counterexample.

b. Is J onto? Prove or give a counterexample.

31. Define F: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+ and G: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+ as follows:

For each (n, m) \in \mathbb{Z}^+ \times \mathbb{Z}^+,

 F(n, m) = 3^n5^m \text{ and } G(n, m) = 3^n6^m 

a. Is F one-to-one? Prove or give a counterexample.

b. Is G one-to-one? Prove or give a counterexample.

a. Is \log_{8}27 = \log_{2}3? Why or why not?

a. Is \log_{16}9 = \log_{4}3? Why or why not?

The properties of logarithm established in 33-35 are used in Sections 11.4 and 11.5.

33. Prove that for all positive real numbers b, x, and y with b \neq 1,

 \log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y 

34. Prove that for all positive real numbers b, x, and y with b \neq 1,

 \log_{b}(xy) = \log_{b}x + \log_{b}y 

35. Prove that for all real numbers a, b, and x with b and x positive and b \neq 1,

 \log_{b}(x^a) = a\log_{b}x 

Exercises 36 and 37 use the following definition: If f: \mathbb{R} \to \mathbb{R} and g: \mathbb{R} \to \mathbb{R} are functions, then the function (f + g): \mathbb{R} \to \mathbb{R} is defined by the formula (f + g)(x) = f(x) + g(x) for every real number x.

36. If f: \mathbb{R} \to \mathbb{R} and g: \mathbb{R} \to \mathbb{R} are both one-to-one, is f + g also one-to-one? Justify your answer.

37. If f: \mathbb{R} \to \mathbb{R} and g: \mathbb{R} \to \mathbb{R} are both onto, is f + g also onto? Justify your answer.

Exercises 38 and 39 use the following definition: If f: \mathbb{R} \to \mathbb{R} and c is a nonzero real number, the function (c \cdot f): \mathbb{R} \to \mathbb{R} is defined by the formula (c \cdot f)(x) = c \cdot (f(x)) for every real number x.

38. Let f: \mathbb{R} \to \mathbb{R} be a function and c a nonzero real number. If f is one-to-one, is c \cdot f also one-to-one? Justify your answer.

39. Let f: \mathbb{R} \to \mathbb{R} be a function and c a nonzero real number. If f is onto, is c \cdot f also onto? Justify your answer.

40. Suppose F: X \to Y is one-to-one.

a. Prove that for every subset A \subseteq X, F^{-1}(F(A)) = A.

b. Prove that for all subsets A_1 and A_2 in X, F(A_1 \cap A_2) = F(A_1) \cap F(A_2).

Let X = \{a, b, c, d, e\} and Y = \{s, t, u, v, w\}. In each of 42 and 43 a one-to-one correspondence F: X \to Y is defined by an arrow diagram. In each case draw an arrow diagram for F^{-1}.

(See page 483 for image.)

(See page 483 for image.)

In 44-55 indicate which of the functions in the referenced exercise are one-to-one correspondences. For each function that is a one-to-one correspondence, find the inverse function.

44. Exercise 10a

45. Exercise 10b

46. Exercise 11a

47. Exercise 11b

48. Exercise 12a

49. Exercise 12b

50. Exercise 21

51. Exercise 22

52. Exercise 15 with the co-domain taken to be the set of all real numbers not equal to 1.

53. Exercise 16 with the co-domain taken to be the set of all real numbers.

54. Exercise 17 with the co-domain taken to be the set of all real numbers not equal to 3

55. Exercise 18 with the co-domain taken to be the set of all real numbers not equal to 1.

56. In Example 7.2.8 a one-to-one correspondence was defined from the power set of \{a, b\} to the set of all strings of $0$'s and $1$'s that have length 2. Thus the elements of these two sets can be matched up exactly, and so the two sets have the same number of elements.

a. Let X = \{x_1, x_2, \dots, x_n\} be a set with n elements. Use Example 7.2.8 as a model to define a one-to-one correspondence from \mathscr{P}(X), the set of all subsets of X, to the set of all strings of $0$'s and $1$'s that have length n.

b. In Section 9.2 we show that there are 2^n strings of 0's and $1$'s that have length n. What does this allow you to conclude about the number of subsets of \mathscr{P}(X)? (This provides an alternative proof of Theorem 6.3.1.)

57. Write a computer algorithm to check whether a function from one finite set to another is one-to-one. Assume the existence of an independent algorithm to compute values of the function.

58. Write a computer algorithm to check whether a function from one finite set to another is onto. Assume the existence of an independent algorithm to compute values of the function.