An Introduction to Logic1
Chapter 1
Teaching Notes, Chapter 1: Some students will have encountered the material on elementary logic earlier in their studies and, if so, this chapter may be safely omitted or quickly reviewed. However, logic is one of the topics that has been eliminated or reduced in coverage in many of the traditional “Mathematics for Teachers” courses. If your students are products of such courses, a careful look Chapter 1 will give them valuable insights into the language of mathematics that will serve them well both in their study of discrete mathematics and in all further mathematical studies. Ensuring an adequate background in the content of Chapter 1 is particularly important if you intend to give careful attention to Chapter 3, An Introduction to Mathematical Proof.
Related Website: An encyclopedic website of sophisticated resources related to the formal study of logic can be found at
Teaching Notes 1.1:
Related Website: At the following web site is found a calculator that will produce a truth table for any symbolic statement:
Exploratory Exercise Set 1.1
1. (a)(b)
p / ~p / ~ (~ p) / p / q / p qT / F / T / T / T / T
F / T / F / T / F / F
F / T / F
F / F / F
(c)
p / q / r / (p q) / (p q) rT / T / T / T / T
T / T / F / T / F
T / F / T / F / F
T / F / F / F / F
F / T / T / F / F
F / T / F / F / F
F / F / T / F / F
F / F / F / F / F
(d)16 cases(e)
1 / 2
2 / 4
3 / 8
4 / 16
5 / 32
n / 2n
- (a)4 (b)16(c)20(d)0(e)16(f)10
Leon / Sarah / Russo / Sue / Sharon
Affirmative
Negative
HarpCollege
SloanCollege / X / X
- (a)Leon and Sue attend the same college; if they attend HarpCollege, then Sarah, Russo, and Sharon must go to SloanCollege as Harp only sent two students to the debate. This is a contradiction as Russo and Sharon attend different schools. Therefore, Leon and Sue must go to SloanCollege.
If Sharon attends SloanCollege, then Russo and Sarah are the two debaters who attend HarpCollege. This is a contradiction as Sarah and Russo represent the same side in the debate. Therefore, Sharon does not attend SloanCollege, which places her at Harp, Russo at Sloan, and Sarah at Harp.
Leon / Sarah / Russo / Sue / SharonAffirmative
Negative
HarpCollege / X / X
SloanCollege / X / X / X
Since Sarah and Sharon are the team from Harp, they must represent opposite sides. Suppose Sarah is the negative debater. Then Sharon is affirmative, Russo is negative, and Sue is negative. If Sarah is a negative debater, then Sharon must be affirmative. This provides three negative debaters, which is a contradiction. Therefore, Sarah must be an affirmative debater, making Sharon a negative debater, and Russo affirmative. Sue and Sharon represent opposite sides, therefore Sue is affirmative, and thus Leon is negative. Therefore, Sharon is the negative debater from HarpCollege selected as outstanding debater.
Leon / Sarah / Russo / Sue / SharonAffirmative / X / X / X
Negative / X / X
HarpCollege / X / X
SloanCollege / X / X / X
(b)We know that the Cats won two games; therefore, they beat the Dogs and the Ants. As the Dogs tied one game, and lost to the Cats, they must have tied the Ants.
The Cats beat the Dogs.
The Cats beat the Ants.
The Dogs tied the Ants.
Since the Ants tied the Dogs, and the Dogs only scored two points, the Ants scored 0, 1, or 2 points against the Dogs. If the Ants scored 0 points against the Dogs, then they scored three points against the Cats, and the Dogs scored two points against the Cats. This is a contradiction, as only one goal was scored against the Cats. If the Ants scored one goal against the Dogs, then the Ants scored two goals against the Cats, again a contradiction. Therefore, the Ants must have scored two goals against the Dogs, and one goal against the Cats. The Dogs scored both of their two points against the Ants in the tie game. Since seven goals were scored against the Ants, two by the Dogs, the Cats must have scored 5 goals against the Ants. Similarly, as the Ants scored two of three goals against the Dogs, the other goal was scored against the Cats.
Therefore, the final score is:
The Cats beat the Dogs 2 to 0.
The Cats beat the Ants 5 to 1.
The Dogs tie the Ants 2 to 2.
4.Answers will vary.
Exercise Set 1.1
1.(a)Statement—False(b)Not a statement(c)Statement—True
(d)Not a statement(e)Not a statement(f)Statement—False
(g)Not a statement(h)Not a statement
2.(a)False(b)True(c)False
(d)True(e)True(f)False
3.(a)False(b)False(c)False
(d)False(e)True(f)False
4.(a)False(b)True(c)True
(d)True(e)True(f)False
5.(a)November has 30 days and Thanksgiving is always on November 25. False
November has 30 days or Thanksgiving is always on November 25. True
(b)The smallest counting number is 2 and 10 is not a multiple of 5. False
The smallest counting number is 2 or 10 is not a multiple of 5. False
(c)2 + 3 = 4 + 1 and 8 6 = 4 12. True
2 + 3 = 4 + 1 or 8 6 = 4 12. True
(d)Triangles are not squares and 3 is smaller than 5. True
Triangles are not squares or 3 is smaller than 5. True
6.(a)AD(b)DA(c)C ~ B(d)~ D ~ A
7.(a)It is snowing and the roofs are not white.
(b)The roofs are not white or the streets are slick.
(c)It is snowing, and either the roofs are white or the streets are not slick.
(d)It is snowing or the streets are slick, and the trees are green.
(e)It is not true that it is snowing and the trees are not green.
(f)It is not true that it is snowing and the streets are slick.
8.(a)True(b)True(c)False
(d)False(e)True(f)False
9.(a)p q(b)~ r(c) ~ q(d)(q r) ~ p
10.(a)(E S) (~ E A)(b)No, they must be enrolled, and in good standing.
(c)Yes, ~ E A is true.
11. (a) (A D) W
(b)Yes. A is true. D is false. W is true. It follows that A D is true so the statement is true.
(c)No. A is false. D is true. It follows that A D is true. However, W is false so the statement is false.
12. (a)If the sign on the second door is true, then the sign on the first door is false, which means that there is not a car behind the door, and there is not a skateboard behind the door. This contradicts the rules of the game. Thus the sign on the second door must be false. This places the car behind the second door.
(b)The sign on the second door describes one of the rules of the game and hence must be true. Hence the sign on the first door is false, placing the car behind the first door.
13. (a)If the husband is a Truthteller, then his statement would be true, thus he is a Liar, a contradiction. Therefore, the husband is a Liar. Since his statement is false, his wife is a Truthteller. Thus, the husband is a Liar, and the wife a Truthteller.
(b)If the wife is a Liar, then her statement is false, making both the giantess and her husband a Truthteller. This is a contradiction, therefore, the wife can not be a Liar. As the wife is a Truthteller, her statement is true, so her husband is a Liar. Thus, the husband is a Liar, and the wife is a Truthteller.
(c)If the first giant is a Truthteller, then the statement “All three of us are Liars” is true, and the first giant is then a Liar. This is a contradiction, therefore the first giant is a Liar. If the third giant is a Truthteller, then the statement “Only one of us is a Liar” is a true statement. As we know the first giant is a Liar, the second Giant must be a Truthteller. Then the statement “Only two of us are Liars.” should be true, but this is a contradiction. Therefore, the third giant is a Liar. Thus, the first giant is a Liar, the second a Truthteller, and the third a Liar.
14.(a)True(b)False(c)False(d)True
15.(a)
p / q / ~ q / p ~ qT / T / F / T
T / F / T / T
F / T / F / F
F / F / T / T
(b)
p / ~ p / p ~ pT / F / F
F / T / F
p / ~ p / ~ (~ p)
T / F / T
F / T / F
16. Since the truth values of p and ~ (~ p) are identical, the two statements convey the same information.
17. (a)The stock market is not bullish and the Dow average is increasing.
(b)The stock market is bullish or the price of utilities is not decreasing.
(c)The stock market is bullish or the price of utilities is decreasing, and the Dow average is increasing.
(d)The price of utilities is not decreasing or the stock market is not bullish, and the Dow average is increasing
18. Let p represent “I invest my money in stocks.” Let q represent the statement “I put my money in a savings account.” Let r represent “I buy gold.”
(a)pq (b) ~ (p q) r (c) ~ r ~ q
- (a) pq(b)pq(c) ~ pq (d) ~ p ~ q
- (a)
p / q / pq / (p q) q
T / T / T / T
T / F / F / F
F / T / F / T
F / F / F / F
(b)
p / q / p q / (p q) pT / T / T / T
T / F / T / T
F / T / T / F
F / F / F / F
21.(a)False(b)False(c)False(d)False
22.(a)True(b)True(c)True(d)False
23.(a)
p / q / r / ~ q / (p ~ q) / (p ~ q) rT / T / T / F / T / T
T / T / F / F / T / F
T / F / T / T / T / T
T / F / F / T / T / F
F / T / T / F / F / F
F / T / F / F / F / F
F / F / T / T / T / T
F / F / F / T / T / F
(b)
p / q / r / (qr) / p (qr)T / T / T / T / T
T / T / F / T / T
T / F / T / T / T
T / F / F / F / F
F / T / T / T / F
F / T / F / T / F
F / F / T / T / F
F / F / F / F / F
(c)
p / q / ~ p / (p ~ p) / (p ~ p) qT / T / F / T / T
T / F / F / T / F
F / T / T / T / T
F / F / T / T / F
(d)
p / q / r / (pq) / ~ (pq) / ~ (pq) rT / T / T / T / F / F
T / T / F / T / F / F
T / F / T / T / F / F
T / F / F / T / F / F
F / T / T / T / F / F
F / T / F / T / F / F
F / F / T / F / T / T
F / F / F / F / T / F
Teaching Notes 1.2: Students can easily learn to show that two statements are equivalent using truth tables without understanding the significance of this fact. One of the most important objectives of this material is to help students understand when one statement can be replaced by another without changing the meaning of the discourse. Exploratory Exercises 1, 2, and 3 together with Exercises 13 through 18 reinforce this objective.
Related Website: Visit for a tour of the mathematics of Venn diagrams including a biography of John Venn.
Exploratory Exercise Set 1.2
1.(a)(1)If Hai is 18, then he can vote.
(2)If the path is muddy, then I will wear my boots.
(3)If I finish studying, then I will go to the party.
(4)If x is a whole number, then x is an integer.
(b)(1)If Hai cannot vote, then he is not 18.
(2)If I do not wear my boots, then the path is not muddy.
(3)If I do not go to the party, then I am not finished studying.
(4)If X is not an integer, then x is not a whole number.
2.(a)Let p represent the statement “Amelia made an A in Mathematics.” Let q represent the statement “Amelia made a B in English.”
(b)Amelia did not make an A in Mathematics and Amelia did not make a B in English.
(c)Ardie cannot pay me now, and Ardie will not work for me later.
3.(a)Let p represent the statement “The month is March.” Let q represent the statement “The azaleas are in bloom.”
(b)The month is March, and the azaleas are not in bloom.
(c)Figure ABCD is a rectangle, and it is not a parallelogram.
4. Answers will vary as students follow different paths to arrive at the information contained in the following chart.
ABC
MPPF
EPPP
SFPF
Exercise Set 1.2
1.(a)AD(b)~ A ~ B(c)~ C ~ A(d)C ~ D
2.(a)True(b)True(c)False(d)True
3.(a)Converse: If one angle of a triangle measures 90°, then the triangle is a right
triangle.
Inverse: If a triangle is not a right triangle, then no angle has a measure of 90°.
Contrapositive: If no angle of a triangle measures 90°, then the triangle is not a
right triangle.
(b)Converse: If a number is odd, then it is prime.
Inverse: If a number is not prime, then it is not odd.
Contrapositive: If a number is not odd, then it is not prime.
(c)Converse: If alternate interior angles are equal, then two lines are parallel.
Inverse: If two lines are not parallel, then alternate interior angles are not equal.
Contrapositive: If alternate interior angles are not equal, then two lines are not
parallel.
(d)Converse: If Joyce is happy, then she is smiling.
Inverse: If Joyce is not smiling, then she is not happy.
Contrapositive: If Joyce is not happy, then she is not smiling.
4.(a)False(b)True(c)True(d)True
5.(a)Converse: If 2 + 2 = 5, then the moon orbits the earth.
Inverse: If the moon does not orbit the earth, then 2 + 2 5.
Contrapositive: If 2 + 2 5, then the moon does not orbit the earth.
(b)Converse: If 2 + 2 = 5, then Lincoln was the first U.S. president.
Inverse: If Lincoln was not the first U.S. president, then 2 + 2 5.
Contrapositive: If 2 + 2 5, then Lincoln was not the first U.S. president.
(c)Converse: If 3 + 3 = 6, then Mexico is the southern neighbor of the United States.
Inverse: If Mexico is not the southern neighbor of the U.S., then 3 + 3 6.
Contrapositive: If 3 + 3 6, then Mexico is not the southern neighbor of the U.S.
(d)Converse: If 3 + 3 = 6, then Canada is in Asia.
Inverse: If Canada is not in Asia, then 3 + 3 6.
Contrapositive: If 3 + 3 6, then Canada is not in Asia.
6.(a)If it is not snowing, then the roofs are white.
(b)If the streets are not slick, then the roofs are not white.
(c)If the roofs are not white and the streets are not slick, then it is snowing.
(d)If it is snowing or the roofs are white, then the streets are not slick.
(e)It is not true that if it is snowing then the roofs are not white.
(f)If the streets are not slick or it is not snowing, then the roofs are not white.
7.If the sign on the second door is true, then the sign on the first door has a false hypothesis and hence is a true statement. This means both signs are true, and this is a contradiction of the rules. Therefore, the first sign must be true, and the second false. This implies the car is behind the first door.
8. (a) If the husband is a Liar, the conditional statement he makes is false. This can only happen if the hypothesis is true. That is, the Giant must be a Truthteller. This is a contradiction. Thus the husband is a Truthteller, so the statement is true, and the wife is a Truthteller.
(b) There are two possibilities: If the husband is a Truthteller, then the statement in the form “p if and only if q” is true, and p is true. Thus q must be true and the wife must be a Truthteller. If the husband is a Liar, then the statement in the form “p if and only if q” is false, and p is false. Thus q must be true, so the wife is a Truthteller. Hence we have no information about the husband, but in either case, the wife is a Truthteller.
9. (a)Converse: If you have fewer cavities, then you brush your teeth with White-as-Snow.
Inverse: If you do not brush your teeth with White-as-Snow, then you do not
have fewer cavities.
Contrapositive: If you do not have fewer cavities, then you do not brush your
teeth with White-as-Snow.
(b)Converse: If you love mathematics, then you like this book.
Inverse: If you do not like this book, then you do not love mathematics.
Contrapositive: If you do not love mathematics, then you do not like this book.
(c)Converse: If you eat Barlies for breakfast, then you are strong.
Inverse: If you are not strong, then you do not eat Barlies for breakfast.
Contrapositive: If you do not eat Barlies for breakfast, then you are not
strong.
(d)Converse: If your clothes are bright and colorful, then you use Wave.
Inverse: If you do not use Wave, then your clothes are not bright and colorful.
Contrapositive: If your clothes are not bright and colorful, then you do not use
Wave.
10.(a)The insured person must pay the $200 difference to Guarantee Auto.
(b)The policy does not state what the insured person must do.
11.(a)The minimum penalty is $75.
(b)The policy statement provides no guidance in this case.
12.(a)If you are careless, then you will have accidents.
(b)If I see June, then my heart throbs.
(c)If two sides of a triangle are congruent, then it is isosceles.
(d)If I pass, then I will be happy.
13.(a)
p / q / ~ q / p q / ~ (p q) / p ~ qT / T / F / T / F / F
T / F / T / F / T / T
F / T / F / T / F / F
F / F / T / T / F / F
Since the last two columns have the same truth values, these statements are equivalent.
(b)It is not true that if Nakisha is a music major, then she sings well.
(c)My dog has fleas, and he does not scratch often.
14.(a)
p / q / ~ p / ~ p q / pqT / T / F / T / T
T / F / F / T / T
F / T / T / T / T
F / F / T / F / F
Since the last two columns have the same truth values, these statements are equivalent.
(b)If I fail English, then I go to summer school.
(c)Either I do not stay up all night, or I will skip the luncheon.
15.(a)It is not true that if Ed drives a truck, then he wears socks.
(b)It is not true that if Ana lives in Memphis, then she does not ride a bus to work.
(c)The black dog sits and he does not want to eat.
(d)The phone rings and trouble does not follow.
16.(a)If I do not eat supper, then I go to bed early.
(b)If Ed does not sleep late, then he is very sluggish.
(c)Either the frog leaves the pond, or he will become lonesome.
(d)Either the cat does not catch a mouse, or she is quite content.
17.(a)
p / q / ~ p / ~ q / pq / ~ (pq) / ~ p ~ qT / T / F / F / T / F / F
T / F / F / T / T / F / F
F / T / T / F / T / F / F
F / F / T / T / F / T / T
Since the last two columns have the same truth values, these statements are equivalent.
(b)
p / q / ~ p / ~ q / pq / ~ (pq) / ~ p ~ qT / T / F / F / T / F / F
T / F / F / T / F / T / T
F / T / T / F / F / T / T
F / F / T / T / F / T / T
Since the last two columns have the same truth values, these statements are equivalent.
(c)I will not take history and I will not take physics.
(d)I am not on the baseball team or I am not on the basketball team.
18.(a)If a triangle is equilateral then it is equiangular and if a triangle is equiangular, then it is equilateral.
(b)A dancer is graceful if and only if he is athletic.
(c)If a mathematician is charming then she is asleep and if a mathematician is asleep, then she is charming.
(d)A connected graph with n vertices is a tree if and only if it has n – 1 edges.
19.(a)If a geometric figure is a triangle, then it is not a square.
(b)If they are birds of a feather, then they flock together.
(c)If a politician is honest, then she/he does not accept bribes.
20.(a)Converse: If a geometric figure is not a square, then it is a triangle.
Contrapositive: If a geometric figure is a square, then it is not a triangle.
(b)Converse: If they flock together, then they are birds of a feather.
Inverse: If they do not flock together, then they are not birds of a feather.
(c)Converse: If a politician does not accept bribes, then he/she is honest.
Inverse: If a politician accepts bribes, then he/she is not honest.
21.(a)
p / q / ~ p / pq / ~ pqT / T / F / T / T
T / F / F / T / T
F / T / T / T / T
F / F / T / F / F
Since the last two columns have the same truth values, these statements are equivalent.
(b)
p / q / ~ p / ~ q / pq / ~ (pq) / ~ p ~ qT / T / F / F / T / F / F
T / F / F / T / F / T / F
F / T / T / F / F / T / F
F / F / T / T / F / T / T
Since these two statements have different truth values in two cases, they are not equivalent.
(c)
p / q / ~ p / ~ q / pq / ~ (pq) / ~ p ~ qT / T / F / F / T / F / F
T / F / F / T / T / F / T
F / T / T / F / T / F / T
F / F / T / T / F / T / T
Since these two statements have different truth values in two cases, they are not equivalent.
(d)
p / q / ~ p / ~ q / pq / ~ p ~ qT / T / F / F / T / T
T / F / F / T / F / T
F / T / T / F / T / F
F / F / T / T / T / T
Since these two statements have different truth values in two cases, they are not equivalent.
22. Number the four statements (1) through (4) in Exercise 22.
Case 1: Dale owns the Plymouth. By (1) the Plymouth is either white or gray. By (2) the Honda is white, so the Plymouth is gray. This means that the Saturn is blue. But then (3) implies that the Plymouth is white. This is a contradiction, so Case I is not possible.