Summary maths teachers guide for freshman course

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UnitONE
Solutions/Answers to Exercises of page 6

1. Which of the following sentences are propositions? For those that are, indicate the truth value.
Solutions/Answers
a. 123 is a prime number. It is a proposition with truth value true T.
b. 0 is an even number. It is a proposition with truth value true T.
c. 𝑥2−4=0. It is not a proposition.
d. Multiply 5𝑥+2 by 3. It is not a proposition.
e. What an impossible question! It is not a proposition.

2. State the negation of each of the following statements.
Solutions/Answers
a. √2 is a rational number. √2 is not a rational number.
b. 0 is not a negative integer. 0 is a negative integer.
c. 111 is a prime number. 111 is not a prime number.

3. Let 𝑝: 15 is an odd number.
𝑞: 21 is a prime number.

Solutions/Answers
a. pq: 15 is an odd number or 21 is a prime number. Truth value True
b. p  q : 15 is an odd number and 21 is a prime number. Truth value False
c. p  q : 15 is not an odd number or 21 is a prime number. Truth value False
d. p  q : 15 is an odd number and 21 is not a prime number. Truth value True
e. p  q : If 15 is an odd number then 21 is a prime number. Truth value False
f. q  p : If 21 is a prime number then 15 is an odd number. Truth value True
a. p  q : If 15 is not an odd number then 21 is not a prime number. Truth value True
g. q  p : If 21 is not a prime number then 15 is not an odd number. Truth value False

4. Complete the following truth table.

p q 𝒒 𝒑𝖠𝒒
T T F F
T F T T
F T F F
F F T F

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Solutions/Answers to Exercises of pages 11-12
1. For statements 𝑝, and 𝑟, use a truth table to show that each of the following pairs of statements is
logically equivalent.
a. (𝑝𝖠𝑞)⟺𝑝 and 𝑝⟹𝑞.
P Q 𝑝𝖠𝑞 (𝑝𝖠𝑞)⟺𝑝 𝑝⟹𝑞 Therefore, (𝑝𝖠𝑞)⟺𝑝 and 𝑝⟹𝑞 are logically equivalent,
T T T T T since the forth and the fifth columns have
T F F F F the same truth values .
F T F T T
F F F T T
b. 𝑝⟹(𝑞𝗏𝑟) and 𝑞⟹(𝑝𝗏𝑟).
P Q r 𝑞𝗏𝑟 P 𝑞 𝑝𝗏𝑟 𝑝⟹(𝑞𝗏𝑟) 𝑞⟹(𝑝𝗏𝑟) Therefore,
T T T T F F T T T 𝑝⟹(𝑞𝗏𝑟) and
T T F T F F F T T 𝑞⟹(𝑝𝗏𝑟) are
T F T T F T T T T logically equivalent.
T F F F F T F F F since the eighth and
F T T T T F T T T the ninth columns have
F T F T T F T T T the same truth values .
F F T T T T T T T
F F F F T T T T T
c. (𝑝𝗏𝑞)⟹𝑟 and (𝑝⟹𝑞)𝖠(𝑞⟹𝑟).
P q r 𝑝𝗏𝑞 𝑝⟹𝑞 𝑞⟹𝑟 (𝑝𝗏𝑞)⟹𝑟 (𝑝⟹𝑞)𝖠(𝑞⟹𝑟) The truth values of the
T T T T T T T T combinations differ in
T T F T T F F F the third row of the
T F T T F T T F seventh and the eighth
T F F T F T F F columns where
(𝑝𝗏𝑞)⟹r is T and
F T T T T T T T
(𝑝⟹𝑞)𝖠(𝑞⟹𝑟) is F so that
F T F T T F F F
𝑝𝗏𝑞)⟹𝑟 and (𝑝⟹𝑞)𝖠(𝑞⟹𝑟) are
F F T F T T T T not logically equivalent.
F F F F T T T T
d. 𝑝⟹(𝑞𝗏𝑟) and (𝑟)⟹(𝑝⟹𝑞).
The truth values of the combinations differ in the second and forth rows of the seventh and
the eighth columns so that 𝑝⟹(𝑞𝗏𝑟) and (𝑟)⟹(𝑝⟹𝑞) are not logically equivalent.
P q r 𝑟 𝑞𝗏𝑟 𝑝⟹𝑞 𝑝⟹(𝑞𝗏𝑟) (𝑟)⟹(𝑝⟹𝑞)
T T T F T T T T
T T F T F T F T
T F T F T F T T
T F F T T F T F
F T T F T T T T
F T F T F T T T
F F T F T T T T
F F F T T T T T


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e. 𝑝⟹(𝑞𝗏𝑟) and ((𝑟)𝖠𝑝)⟹𝑞.
p q r 𝑟 𝑞𝗏𝑟 (𝑟)𝖠𝑝 𝑝⟹(𝑞𝗏𝑟) ((𝑟)𝖠𝑝)⟹𝑞 The truth values of the
T T T F T F T T combinations the propositions
T T F T T T T T of the seventh and
the eighth columns are
T F T F T F T T the same so that
T F F T F T F F 𝑝⟹(𝑞𝗏𝑟) and (𝑟)⟹(𝑝⟹𝑞)
F T T F T F T T are logically equivalent.
F T F T T F T T
F F T F T F T T
F F F T F F T T

2. For statements 𝑝, q, and 𝑟, show that the following compound statements are tautology.
a. 𝑝⟹(𝑝𝗏𝑞).
p q 𝑝𝗏q 𝑝⟹(𝑝𝗏𝑞) 𝑝⟹(𝑝𝗏𝑞 is a tautology since all the possible combinations
T T T T in the last column are all T
T F T T
F T T T
F F F T

b. (𝑝𝖠(𝑝⟹𝑞))⟹𝑞.
p q 𝑝⟹q 𝑝𝖠(𝑝⟹𝑞) (𝑝𝖠(𝑝⟹𝑞))⟹𝑞 (𝑝𝖠(𝑝⟹𝑞))⟹𝑞 is a tautology since all
T T T T T the possible combinations in the last
T F F F T column are all T
F T T F T
F F T F T

c. ((𝑝⟹𝑞)𝖠(𝑞⟹𝑟))⟹(𝑝⟹𝑟).
p q r 𝑝⟹q 𝑞⟹𝑟 𝑝⟹𝑟 (𝑝⟹𝑞)𝖠(𝑞⟹𝑟) ((𝑝⟹𝑞)𝖠(𝑞⟹𝑟))⟹(𝑝⟹𝑟)
T T T T T T T T
T T F T F F F T
T F T F T T F T
T F F F T F F T
F T T T T T T T
F T F T F T F T
F F T T T T T T
F F F T T T T T

((𝑝⟹𝑞)𝖠(𝑞⟹𝑟))⟹(𝑝⟹𝑟) is a tautology since all the possible combinations in the last
column are all T

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3. For statements 𝑝 and 𝑞, show that (𝑝𝖠𝑞)𝖠(𝑝𝖠𝑞)
is a contradiction.

P q 𝑞 𝑝𝖠q 𝑝𝖠𝑞 (𝑝𝖠𝑞)𝖠(𝑝𝖠𝑞)
T T F F T F (𝑝𝖠𝑞)𝖠(𝑝𝖠𝑞) is a contradiction since the last
T F T T F F column has all the truth values F
F T F F F F
F F T F F F

4. Write the contrapositive and the converse of the following conditional statements.
a. If it is cold, then the lake is frozen.
Contrapositive: If the lake is not frozen then it is not cold.
Converse: If the lake is frozen then it is cold.
b. If Solomon is healthy, then he is happy.
Contrapositive: If he is not happy then Solomon is not healthy.
Converse: If he is happy then Solomon is healthy.
c. If it rains, Tigist does not take a walk.
Contrapositive: If Tigist takes a walk, it doesn't rain.
Converse: If Tigist doesn't take a walk, it rains.

5. Let 𝑝 and 𝑞 be statements. Which of the following implies that 𝑝𝗏𝑞 is false?

𝑝𝗏𝑞 is false means p is True and q is False so that:

a. 𝑝𝗏𝑞 is false.

𝑝 is F and 𝑞 is T which implies F 𝗏 T which implies T so that 𝑝𝗏𝑞 is T

b. 𝑝𝗏𝑞 is true.

This means F or F which is F

c. 𝑝𝖠𝑞 is true.

This means F 𝖠 T which is F

d. 𝑝⟹𝑞 is true.

This means T implies F which is F

e. 𝑝𝖠𝑞 is false.

This means T and F which is F

So for in general for question number 5 , the answers are b, c, d, e



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