Solutions to Susanna Epp Discrete Math 5th Edition
1 Chapter 1
1.1 Exercise Set 1.1
In each of 1–6, fill in the blanks using a variable or variables to rewrite the
given statement.
1.1.1 Problem 1
Is there a real number whose square is →1?
(a) Is there a real number x such that ?
Proof. Is there a real number x such that x2 = →1?
(b) Does there exist such that x2 = →1?
Proof. Does there exist a real number x such that x2 = →1?
1.1.2 Problem 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?
Note: There are integers with this property. Can you think of one?
(a) Is there an integer n such that n has ?
Proof. Is there an integer n such that n has a remainder of 2 when it is divided by 5
and a remainder of 3 when it is divided by 6?
(b) Does there exist such that if n is divided by 5 the remainder is 2 and if ?
Proof. Does there exist an integer n such that if n is divided by 5 the remainder is 2
and if n is divided by 6 the remainder is 3?
1
,1.1.3 Problem 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
.
Proof. Given any two distinct real numbers a and b, there is a real number c such that
c is between a and b.
(b) For any two , such that c is between a and b.
Proof. For any two distinct real numbers a and b, there exists a real number c such that
c is between a and b.
1.1.4 Problem 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 .
Proof. Given any real number r, there is a real number s such that s is greater than r.
(b) For any , such that s > r.
Proof. For any real number r, there exists a real number s such that s > r.
1.1.5 Problem 5
The reciprocal of any positive real number is positive.
(a) Given any positive real number r, the reciprocal of .
Proof. Given any positive real number r, the reciprocal of r is positive.
(b) For any real number r, if r is , then .
Proof. For any real number r, if r is positive, then 1/r is positive.
(c) If a real number r , then .
Proof. If a real number r is positive, then 1/r is positive.
2
,1.1.6 Problem 6
The cube root of any negative real number is negative.
(a) Given any negative real number s, the cube root of .
Proof. Given any negative real number s, the cube root of s is negative.
(b) For any real number s, if s is , then .
→
Proof. For any real number s, if s is negative, then 3
s is negative.
(c) If a real number s , then .
→
Proof. If a real number s is negative, then 3
s is negative.
1.1.7 Problem 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.
Proof. Rewrite: There are real numbers such that their sum is less than their di!erence.
True: 0 and ↑1 have this property: ↑1 = 0 + (↑1) < 0 ↑ (↑1) = 1
(b) There is a real number x such that x2 < x.
Proof. Rewrite: there is a real number whose square is less than itself.
! "2
1 1 1
True: 1/2 has this property: = <
4 2 2
(c) For every positive integer n, n2 ↓ n.
Proof. Rewrite: The square of every positive integer is greater than or equal to itself.
True: if we look at the first few examples it holds: 12 = 1 ↓ 1, 22 = 4 ↓ 2, 32 = 9 ↓ 3
and so on. This is however not a proof. Later we’ll learn methods to prove this for all
positive integers.
(d) For all real numbers a and b, |a + b| ↔ |a| + |b|.
Proof. Rewrite: for all two real numbers, the absolute value of their sum is less than or
equal to the sum of their absolute values.
True: this is known as the Triangle Inequality and it will be proved later.
In each of 8-13, fill in the blanks to rewrite the given statement.
3
, 1.1.8 Problem 8
For every object J, if J is a square then J has four sides.
(a) All squares .
Proof. All squares have four sides.
(b) Every square .
Proof. Every square has four sides.
(c) If an object is a square, then it .
Proof. If an object is a square, then it has four sides.
(d) If J , then J .
Proof. If J is a square, then J has four sides.
(e) For every square J, .
Proof. For every square J, J has four sides.
1.1.9 Problem 9
For every equation E, if E is quadratic then E has at most two real solutions.
(a) All quadratic equations .
Proof. All quadratic equations have at most two real solutions.
(b) Every quadratic equation .
Proof. Every quadratic equation has at most two real solutions.
(c) If an equation is quadratic, then it .
Proof. If an equation is quadratic, then it has at most two real solutions.
(d) If E , then E .
Proof. If E is a quadratic equation, then E has at most two real solutions.
(e) For every quadratic equation E, .
Proof. For every quadratic equation E, E has at most two real solutions.
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1 Chapter 1
1.1 Exercise Set 1.1
In each of 1–6, fill in the blanks using a variable or variables to rewrite the
given statement.
1.1.1 Problem 1
Is there a real number whose square is →1?
(a) Is there a real number x such that ?
Proof. Is there a real number x such that x2 = →1?
(b) Does there exist such that x2 = →1?
Proof. Does there exist a real number x such that x2 = →1?
1.1.2 Problem 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?
Note: There are integers with this property. Can you think of one?
(a) Is there an integer n such that n has ?
Proof. Is there an integer n such that n has a remainder of 2 when it is divided by 5
and a remainder of 3 when it is divided by 6?
(b) Does there exist such that if n is divided by 5 the remainder is 2 and if ?
Proof. Does there exist an integer n such that if n is divided by 5 the remainder is 2
and if n is divided by 6 the remainder is 3?
1
,1.1.3 Problem 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
.
Proof. Given any two distinct real numbers a and b, there is a real number c such that
c is between a and b.
(b) For any two , such that c is between a and b.
Proof. For any two distinct real numbers a and b, there exists a real number c such that
c is between a and b.
1.1.4 Problem 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 .
Proof. Given any real number r, there is a real number s such that s is greater than r.
(b) For any , such that s > r.
Proof. For any real number r, there exists a real number s such that s > r.
1.1.5 Problem 5
The reciprocal of any positive real number is positive.
(a) Given any positive real number r, the reciprocal of .
Proof. Given any positive real number r, the reciprocal of r is positive.
(b) For any real number r, if r is , then .
Proof. For any real number r, if r is positive, then 1/r is positive.
(c) If a real number r , then .
Proof. If a real number r is positive, then 1/r is positive.
2
,1.1.6 Problem 6
The cube root of any negative real number is negative.
(a) Given any negative real number s, the cube root of .
Proof. Given any negative real number s, the cube root of s is negative.
(b) For any real number s, if s is , then .
→
Proof. For any real number s, if s is negative, then 3
s is negative.
(c) If a real number s , then .
→
Proof. If a real number s is negative, then 3
s is negative.
1.1.7 Problem 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.
Proof. Rewrite: There are real numbers such that their sum is less than their di!erence.
True: 0 and ↑1 have this property: ↑1 = 0 + (↑1) < 0 ↑ (↑1) = 1
(b) There is a real number x such that x2 < x.
Proof. Rewrite: there is a real number whose square is less than itself.
! "2
1 1 1
True: 1/2 has this property: = <
4 2 2
(c) For every positive integer n, n2 ↓ n.
Proof. Rewrite: The square of every positive integer is greater than or equal to itself.
True: if we look at the first few examples it holds: 12 = 1 ↓ 1, 22 = 4 ↓ 2, 32 = 9 ↓ 3
and so on. This is however not a proof. Later we’ll learn methods to prove this for all
positive integers.
(d) For all real numbers a and b, |a + b| ↔ |a| + |b|.
Proof. Rewrite: for all two real numbers, the absolute value of their sum is less than or
equal to the sum of their absolute values.
True: this is known as the Triangle Inequality and it will be proved later.
In each of 8-13, fill in the blanks to rewrite the given statement.
3
, 1.1.8 Problem 8
For every object J, if J is a square then J has four sides.
(a) All squares .
Proof. All squares have four sides.
(b) Every square .
Proof. Every square has four sides.
(c) If an object is a square, then it .
Proof. If an object is a square, then it has four sides.
(d) If J , then J .
Proof. If J is a square, then J has four sides.
(e) For every square J, .
Proof. For every square J, J has four sides.
1.1.9 Problem 9
For every equation E, if E is quadratic then E has at most two real solutions.
(a) All quadratic equations .
Proof. All quadratic equations have at most two real solutions.
(b) Every quadratic equation .
Proof. Every quadratic equation has at most two real solutions.
(c) If an equation is quadratic, then it .
Proof. If an equation is quadratic, then it has at most two real solutions.
(d) If E , then E .
Proof. If E is a quadratic equation, then E has at most two real solutions.
(e) For every quadratic equation E, .
Proof. For every quadratic equation E, E has at most two real solutions.
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