discrete_mathematics_with_applications/chapter_1/exercises.md

Exercise Set 1.1

Page 28

In each of 1-6, fill in the blanks using a variable or variables to rewrite the given statement.

1. Is there a real number whose square is -1?

a. Is there a real number x such that ______?

b. Does there exist ______ such that x^2 = -1?

Solution

a. Is there a real number x such that x^2 = -1?

b. Does there exist a real number x such that x^2 = -1?

2. Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6?

a. Is there an integer n such that n has ______?

b. Does there exist ______ such that if n is divided by 5 the remainder is 2 and if ______?

Note: There are integers with this property. Can you think of one?

Solution

a. Is there an integer n such that n has a remainder of 2 when n is divided by 5 and a remainder of 3 when n is divided by 6?

b. Does there exist a number n such that if n is divided by 5 the remainder is 2 and if n is divided by 6 the remainder is 3?

Note: There are integers with this property. Can you think of one?

 27 \mod 5 = 2 

 27 \mod 6 = 3 

3. Given any two distinct real numbers, there is a real number in between them.

a. Given any two distinct real numbers a and b, there is a real number c such that c is ______.

b. For any two ______, ______ such that c is between a and b.

Solution

a. Given any two distinct real numbers a and b, there is a real number c such that c is a \leq c \leq b.

b. For any two distinct real numbers a and b, there exists a real number c such that c is between a and b.

4. Given any real number, there is a real number that is greater.

a. Given any real number r, there is ______ s such that s is ______

b. For any ______, ______ such that s > r.

Solution

a. Given any real number r, there is a real number s such that s is greater than r.

b. For any real number r, there exists a real number s such that s > r.

5. The reciprocal of any positive real number is positive.

a. Given any positive real number r, the reciprocal of ______.

b. For any real number , if r is ______, then ______.

c. If a real number r ______, then ______.

Solution

a. Given any positive real number r, the reciprocal of r is positive.

b. For any real number r, if r is positive, then the reciprocal of r is positive.

c. If a real number r is positive, then the reciprocal of r is positive.

6. The cube root of any negative real number is negative.

a. Given any negative real number s, the cube root of ______.

b. For any real number s, if s is ______, then ______.

c. If a real number s ______, then ______.

Solution

a. Given any negative real number s, the cube root of s is negative.

b. For any real number s, if s is negative, then the cube root of s is negative.

c. If a real number s is negative, then the cube root of s is negative.

7. Rewrite the following statements less formally, without using variables. Determine, as best as you can, whether the statements are true or false.

a. There are real numbers u and v with the property that u + v < u - v.

b. There is a real number x such that x^2 < x.

c. For every positive integer n, n^2 \geq n.

d. For all real numbers a and b, |a + b| \leq |a| + |b|.

Solution

a. There are real numbers u and v with the property that u + v < u - v.

There are two distinct real numbers where the sum of those two numbers is less than the difference of those two numbers.

This is true if you consider our domain is all real numbers which include negatives. For example:

 1 + (-1) = 0 

 1 - (-1) = 2 

 0 < 2 

b. There is a real number x such that x^2 < x.

There is a real number which is greater than it's square.

This is true for any fraction/decimal. Consider:

 \left(\frac{1}{4}\right)^2 = \frac{1}{16} 

 \frac{1}{16} < \frac{1}{4} 

c. For every positive integer n, n^2 \geq n.

For all positive integers, an integer's square is always greater than or equal to the integer.

This is true. Starting at 1 we get 1^2 \geq 1, which is true, 2^2 \geq 2 is true, and so on. We're essentially multiplying each side of the inequality by some positive integer, which we know from algebra does not change the direction of the inequality, so this statement holds true.

d. For all real numbers a and b, |a + b| \leq |a| + |b|.

For any two distinct real numbers, the absolute value of their sum is less than or equal to the sum of the absolute values of each number.

This is true, if both a and b are positive numbers or both a and b are negative integers, then the two statements are equal. If either a or b is negative and the other is positive, then the left statement will always be less than the right hand statement.

---

In each of 8-13, fill in the blanks to rewrite the given statement.

8. For every object J, if J is a square then J has four sides.

a. All squares ______.

b. Every square ______.

c. If an object is a square, then it ______.

d. If J ______, then J ______.

e. For every square J, ______.

Solution

a. All squares have four sides.

b. Every square has four sides.

c. If an object is a square, then it has four sides.

d. If J is a square, then J has four sides.

e. For every square J, J has four sides.

9. For every equation E, if E is quadratic then E has at most two real solutions.

a. All quadratic equations ______.

b. Every quadratic equation ______.

c. If an equation is quadratic, then it ______.

d. If E ______, then E ______.

e. For every quadratic equation E, ______.

Solution

a. All quadratic equations have at most two real solutions.

b. Every quadratic equation has at most two real solutions.

c. If an equation is quadratic, then it has at most two real solutions.

d. If E is a quadratic equation, then E has at most two real solutions.

e. For every quadratic equation E, E has at most two real solutions.

10. Every nonzero real number has a reciprocal.

a. All nonzero real numbers ______.

b. For every nonzero real number r, there is ______ for r.

c. For every nonzero real number r, there is a real number s such that ______.

Solution

a. All nonzero real numbers have reciprocals.

b. For every nonzero real number r, there is a reciprocal for r.

c. For every nonzero real number r, there is a real number s such that s is a reciprocal of r.

11. Every positive number has a positive square root.

a. All positive numbers ______.

b. For every positive number e, there is ______ for e.

c. For every positive number e, there is a positive number r such that ______.

Solution

a. All positive numbers have positive square roots.

b. For every positive number e, there is a positive square root for e.

c. For every positive number e, there is a positive number r such that r is a positive square root for e.

12. There is a real number whose product with every number leaves the number unchanged.

a. Some ______ has the property that its ______.

b. There is a real number r such that the product of r ______.

c. There is a real number r with the property that for every real number s, ______.

Solution

a. Some real number has the property that its product with every number leaves the number unchanged.

b. There is a real number r such that the product of r with every number leaves r unchanged.

c. There is a real number r with the property that for every real number s, such that rs = s.

13. There is a real number whose product with every real number equals zero.

a. Some _____ has the property that its ______.

b. There is a real number a such that the product of a ______.

c. There is a real number a with the property that for every real number b, ______.

Solution

a. Some real number has the property that its product with every real number equals zero.

b. There is a real number a such that the product of a with every real number equals zero.

c. There is a real number a with the property that for every real number b, ab = 0.

---

Exercise Set 1.2

Page 37

1. Which of the following sets are equal?

 A = \{a, b, c, d\} 

 B = \{d, e, a, c\} 

 C = \{d, b, a, c\} 

 D = \{a, a, d, e, c, e\} 

Solution

 A = C 

 B = D 

2. Write in words how to read each of the following out loud.

a. \{x \in \mathbb{R}^+ | 0 < x < 1\}

b. \{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}

c. \{n \in \mathbb{Z} | n \text{ is a factor of } 6\}

d. \{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}

Solution

a. \{x \in \mathbb{R}^+ | 0 < x < 1\}

The set of all positive real numbers x such that x is greater than 0 and less than 1.

b. \{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}

The set of all real numbers x such that x is less than or equal to 0 or x is greater than or equal to 1.

c. \{n \in \mathbb{Z} | n \text{ is a factor of } 6\}

The set of all integers n such that n is a factor of 6.

d. \{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}

The set of all positive integers n such that n is a factor of 6.

a. Is 4 = \{4\}?

b. How many elements are in the set \{3, 4, 3, 5\}?

c. How many elements are in the set \{1, \{1\}, \{1, \{1\}\}\} ?

Solution

a. Is 4 = \{4\}?

No, the symbol 4, which represents the number four, does not equal the set that contains an element that is the number 4.

b. How many elements are in the set \{3, 4, 3, 5\}?

There are 3 elements in the set \{3, 4, 3, 5\}. Repeated elements are not counted as more than 1 element in a set.

c. How many elements are in the set \{1, \{1\}, \{1, \{1\}\}\} ?

There are three elements in the set, namely the symbol 1, the set \{1\}, and the set \{1, \{1\}\}.

a. Is 2 \in \{2} ?

b. How many elements are in the set \{2, 2, 2, 2\} ?

c. How many elements are in the set \{0, \{0\}\} ?

d. Is \{0\} \in \{\{0\}, \{1\}\} ?

e. Is 0 \in \{\{0\}, \{1\}\} ?

Solution

a. Is 2 \in \{2} ?

No, the symbol 2 which represents the number two, is not equal to the set \{2\}, which is a set that contains the element 2.

b. How many elements are in the set \{2, 2, 2, 2\} ?

There is one element in the set \{2, 2, 2, 2\}, namely the element 2.

c. How many elements are in the set \{0, \{0\}\} ?

There are two elements in the set, namely the symbol 0, and the set \{0\}.

d. Is \{0\} \in \{\{0\}, \{1\}\} ?

Yes, the set of \{0\} is in the set \{\{0\}, \{1\}\}, as the set contains both the sets \{0\} and \{1\}.

e. Is 0 \in \{\{0\}, \{1\}\} ?

No, the symbol 0, representing the number zero, is not in the set, which holds two sets with the symbols in them.

5. Which of the following sets are equal?

A = {0, 1, 2} \ B = {x \in \mathbb{R} | -1 \leq x < 3} \ C = {x \in \mathbb{R} | -1 < x < 3} \ D = {x \in \mathbb{Z} | -1 < x < 3} \ E = {x \in \mathbb{Z}^+ | -1 < x < 3}

Solution

None of these sets are equal. A = E might have worked had A not included 0, but E essentially evaluates to E = \{1, 2\}, and does not include 0.

6. For each integer n, let T_n = \{n, n^2\}. How many elements are in each of T_2, T_{-3}, T_1, and T_0? Justify your answers.

Solution

 T_2 = \{2, 2^2\} = \{2, 4\} \quad \text{ two elements } 

 T_{-3} = \{-3, (-3)^2\} = \{-3, 9\} \quad \text{ two elements }

 T_1  = \{1, 1^2\} = \{1, 1\} = \{1\} \quad \text{ one element } 

 T_0 = \{0, 0^2\} = \{0, 0\} = \{0\} \quad \text{ one element } 

7. Use the set-roster notation to indicate the elements in each of the following sets.

a. S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}

b. T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}

c. U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}

d. V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}

e. W = \{t \in \mathbb{Z} | 1 < t < -3\}

f. X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}

Solution

a. S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}

 \{-1, 1\} 

b. T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}

 \{0, 2\} 

c. U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}

 \emptyset 

d. V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}

 \mathbb{Z} 

e. W = \{t \in \mathbb{Z} | 1 < t < -3\}

 \emptyset 

f. X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}

 \mathbb{Z}^+ 

8. Let A = \{c, d, f, g\}, B = \{f, j\}, and C = \{d, g\}. Answer each of the following questions. Give reasons for your answers.

a. Is B \subseteq A?

b. Is C \subseteq A?

c. Is C \subseteq C?

d. Is C a proper subset of A?

Solution

a. Is B \subseteq A?

No, because every element of B must be an element of A by definition of a subset, but j \in B, but j \notin A.

b. Is C \subseteq A?

Yes, every element of C is an element of A.

c. Is C \subseteq C?

Yes, every element of C is an element of C. By implication, every set is a subset of itself.

d. Is C a proper subset of A?

Yes, C \subset A, but C \neq A. Every element of C is an element of A, but C does not equal A, which is the definition of a proper subset.

a. Is 3 \in \{1, 2, 3\}?

b. Is 1 \subseteq \{1}?

c. Is \{2\} \in \{1, 2\}?

d. Is \{3\} \in \{1, \{2\}, \{3\}\}?

e. Is 1 \in \{1\}?

f. Is \{2\} \subseteq \{1, \{2\}, \{3\}\}?

g. Is \{1\} \subseteq \{1, 2\}?

h. Is 1 \in \{\{1\}, 2\}?

i. Is \{1\} \subseteq \{1, \{2\}\}?

j. Is \{1\} \subseteq \{1\}?

Solution

a. Is 3 \in \{1, 2, 3\}?

Yes, the symbol 3, representing the number three, is in the set \{1, 2, 3\}.

b. Is 1 \subseteq \{1}?

No, the number 1 is not a set, and therefore cannot be a subset of \{1\}.

c. Is \{2\} \in \{1, 2\}?

No, the subset \{2\} is not in the set \{1, 2\}, the number 2 is in the subset, but not the set \{2\}.

d. Is \{3\} \in \{1, \{2\}, \{3\}\}?

Yes, the set \{3\} is an element of \{1, \{2\}, \{3\}\}.

e. Is 1 \in \{1\}?

Yes, the number 1 is in the set \{1\}.

f. Is \{2\} \subseteq \{1, \{2\}, \{3\}\}?

No, the set \{2\} holds the element 2, and 2 is not an element in \{1, \{2\}, \{3\}\}.

g. Is \{1\} \subseteq \{1, 2\}?

Yes, the set \{\1} holds the element 1, and 1 is an element of \{1, 2\}.

h. Is 1 \in \{\{1\}, 2\}?

No, the element 1 is not in \{\{1\}, 2\}.

i. Is \{1\} \subseteq \{1, \{2\}\}?

Yes, the set \{1\} holds the element 1, which is an element of \{1, \{2\}\}.

j. Is \{1\} \subseteq \{1\}?

Yes \{1\} holds the element 1, which is an element of \{1\}. They are equal and it is implied that any set is a subset of itself.

a. Is ((-2)^2, -2^2) = (-2^2, (-2)^2)?

b. Is (5, -5) = (-5, 5)?

c. Is (8 - 9, \sqrt[3]{-1}) = (-1, -1)?

d. Is \left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\dfrac{3}{6}, -8\right)

Solution

a. Is ((-2)^2, -2^2) = (-2^2, (-2)^2)?

 ((-2)^2, -2^2) = (4, -4) \neq (-4, 4) = (-2^2, (-2)^2) 

So no, they are not equal. For ordered pair tuples to be equal, the order matters and so each entry into the tuple must match the other for them to be equal.

b. Is (5, -5) = (-5, 5)?

No, ordered pair tuples require that the entries be equal to each other in order.

c. Is (8 - 9, \sqrt[3]{-1}) = (-1, -1)?

 (8 - 9, \sqrt[3]{-1}) = (-1, -1) = (-1, -1) 

So yes, these two ordered pair tuples are equal.

d. Is \left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\dfrac{3}{6}, -8\right)

 \left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\frac{1}{2}, -8\right) = \left(\frac{1}{2}, -8\right) = \left(\dfrac{3}{6}, -8\right) 

So yes, these two ordered pair tuples are equal.

11. Let A = \{w, x, y, z\} and B = \{a, b\}. Use set-roster notation to write each of the following sets, and indicate the number of elements that are in each set.

a. A \times B

b. B \times A

c. A \times A

d. B \times B

Solution

a. A \times B

 A \times B = \{(w, a), (w, b), (x, a), (x, b), (y, a), (y, b), (z, a), (z, b)\} 

There are 8 elements in A \times B.

b. B \times A

 B \times A = \{(a, w), (b, w), (a, x), (b, x), (a, y), (b, y), (a, z), (b, z)\} 

There are 8 elements in B \times A.

c. A \times A

 A \times A = \{(w, w), (w, x), (w, y), (w, z), (x, w), (x, x), (x, y), (x, z), (y, w), (y, x), (y, y), (y, z), (z, w), (z, x), (z, y), (z, z)\} 

There are 16 elements in A \times A.

d. B \times B

 B \times B = \{(a, a), (a, b), (b, a), (b, b)\} 

There are 4 elements in B \times B.

12. Let S = \{2, 4, 6\} and T = \{1, 3, 5\}. Use the set-roster notation to write each of the following sets, and indicate the number of elements that are in each set.

a. S \times T

b. T \times S

c. S \times S

d. T \times T

Solution

a. S \times T

 S \times T = \{(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)\} 

There are 9 elements in S \times T.

b. T \times S

 T \times S = \{(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)\} 

There are 9 elements in T \times S.

c. S \times S

 S \times S = \{(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)\} 

There are 9 elements in S \times S.

d. T \times T

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

There are 9 elements in T \times T.

13. Let A = \{1, 2, 3\}, B = \{u\}, and C = \{m, n\}. Find each of the following sets.

a. A \times (B \times C)

b. (A \times B) \times C

c. A \times B \times C

Solution

a. A \times (B \times C)

 B \times C = \{(u, m), (u, n)\} 

 A \times (B \times C) = \{(1, (u, m)), (1, (u, n)), (2, (u, m)), (2, (u, n)), (3, (u, m)), (3, (u, n))\} 

b. (A \times B) \times C

 A \times B = \{(1, u), (2, u), (3, u)\} 

 (A \times B) \times C = \{((1, u), m), ((1, u), n), ((2, u), m), ((2, u), n), ((3, u), m), ((3, u), n)\} 

c. A \times B \times C

 A \times B \times C = \{(1, u, m), (1, u, n), (2, u, m), (2, u, n), (3, u, m), (3, u, n)\} 

14. Let R = \{a\}, S = \{x, y\}, and T = \{p, q, r\}. Find each of the following sets.

a. R \times (S \times T)

 S \times T = \{(x, p), (x, q), (x, r), (y, p), (y, q), (y, r)\} 

 R \times (S \times T) = \{(a, (x, p)), (a, (x, q)), (a, (x, r)), (a, (y, p)), (a, (y, q)), (a, (y, r))\} 

b. (R \times S) \times T

 R \times S = \{(a, x), (a, y)\} 

 (R \times S) \times T = \{((a, x), p), ((a, x), q), ((a, x), r), ((a, y), p), ((a, y), q), ((a, y), r)\} 

c. R \times S \times T

Solution

 R \times S \times T = \{(a, x, p), (a, x, q), (a, x, r), (a, y, p), (a, y, q), (a, y, r)\} 

a. R \times (S \times T)

b. (R \times S) \times T

c. R \times S \times T

15. Let S = \{0, 1\}. List all the strings of length 4 over S that contain three or more $0$'s.

Solution

0000, 0001, 0010, 0100, 1000

16. Let T = \{x, y\}. List all the strings of length 5 over T that have exactly one y.

Solution

xxxxy, xxxyx, xxyxx, xyxxx, yxxxx