Section 1 Truth TablesRecall that a statement is a group of words or symbols that can be classified collec-tively as true or false. The claim “ ” is a true statement, whereas “A pentagon has six sides” is a false statement. However, “Man overboard!” is not astatement at all. Symbolically, we represent statements by letters such as P, Q, and R.The negation of a given statement P is written ; when P is true, is false, and viceversa. This relationship between a statement and its negation can be expressed by atruth table. The first line of the truth table shown indicates “If P is true, then isfalse.” The second line in the table indicates “When P is false, is true.” Supposethat P is the true statement “Abraham Lincoln lived in Illinois.” Then “AbrahamLincoln did not live in Illinois” is false.

A truth table is a valuable tool for examining the truth of a statement that is morecomplex. What exactly is a truth table?

CONJUNCTIONStatements can be combined to form compound statements. For example, a state-ment of the form “P and Q” is called the conjunction of P and Q. In symbols, the con-junction is written . For the conjunction to be true, it is necessary for P to betrue and for Q to be true. If either statement is false, the conjunction is false. To allowfor all possible true/false combinations, four rows are needed in the truth table of theconjunction. When P is true, Q may be true or false (two rows of the table). When P isfalse, Q may be true or false (two additional rows).

P � Q

DEFINITION: A truth table is a table that provides the truth values of astatement by considering all possible true/false combinations of the state-ment’s components.

�P ��P

�P

�P�P

5 � 7 � 12

Logic Appendix

P

T FF T

�P

P Q

T T TT F FF T FF F F

P � Q

EXAMPLE 1

Determine which of the following are statements. For each one that is a state-ment, is it true or false?

a) c) Are you Mike?b) Babe Ruth played baseball. d) , if P is true.

Solutiona) False statement c) Not a statementb) True statement d) False statement

�P4 � 3 � 5

A1

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DISJUNCTIONA compound statement of the form “P or Q” is called the disjunction of P and Q. Insymbols, the disjunction is written . For the disjunction to be true, either P istrue, Q is true, or both P and Q are true. A disjunction is false only if P and Q are bothfalse. To understand the first line in the truth table, consider this situation: You canjoin the Math Club if you have an A average or you are enrolled in a mathematicsclass. If you satisfy both requirements, you may still join the club.

In some cases, we use parentheses to clarify the meaning of a compound state-ment. In this context, parentheses are given priority just as they are in numericalexpressions. See Example 4.

IMPLICATIONThe final compound statement that we consider is of the form “If P, then Q.” Thisstatement is called an implication or a conditional statement. In symbols, we write

P � Q

A2 Logic Appendix

EXAMPLE 3

Let “Babe Ruth played baseball” and Classify as true orfalse:

a) b)

Solutiona) The disjunction is true because P is true even though Q is false.b) The disjunction is true because P is true and �Q is also true.

P � �QP � Q

Q � “4 � 3 � 5.”P �

EXAMPLE 4

Where P, Q, and R are statements, suppose that P is true, Q is false, and R is false.Classify the statement as true or false.

SolutionThe statement in parentheses is a disjunction of the form “T or F” and is true.Then is a conjunction of the form “T and T,” so the given state-ment is true.

NOTE: The steps used to determine the truth of the statement in Example 4 canbe written as follows:

TT � (T)

T � (T � F)P � (�Q � R)

P � (�Q � R)

P � (�Q � R)

P Q

T T TT F TF T TF F F

P � Q

EXAMPLE 2

Let “Babe Ruth played baseball” and Classify as true or false:

a) b)

Solutiona) The conjunction is false because Q is false.b) The conjunction is true because P is true and is also true.�Q

P � �QP � Q

Q � “4 � 3 � 5.”P �

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Section 1 Truth Tables A3

. The conditional statement makes a promise and fails to satisfy the conditionsof this promise only when P is true and Q is false (see the truth table). Consider theclaim “If you are good, then I’ll give you a dollar.” The only way the claim is false iswhen “you are good, but I don’t give you the dollar.”

In , P is called the antecedent and Q is called the consequent.Truth tables have several applications. They are used to show that:

1. Two statements are logically equivalent, meaning their truth values are thesame.

2. The negation of a compound statement has a particular form. For instance, wecan show that the negation of is .

3. Some statements are always true. Such statements are called tautologies.4. An argument is valid. Valid arguments are discussed in Section 2.

Recall that the conditional statement “If P, then Q” is expressed symbolically by. Related to the conditional are its

Converse If Q, then P.Inverse If not P, then not Q.Contrapositive If not Q, then not P.

LOGICAL EQUIVALENCE OF STATEMENTS

In Example 5, we will show that an implication and its contrapositive are logi-cally equivalent. Because the example involves two simple statements P and Q, thetruth table shown has four horizontal rows. At the top of the six vertical columns ofthe truth table are P and Q as well as certain statements that involve statement P,statement Q, or both. Because our goal is to compare the truth values of and

, the column headings of the truth table are chosen accordingly.

A double arrow is used to show that two statements are logically equivalent. InExample 5, .(P → Q) ↔ (�Q → �P)

�Q → �PP → Q

DEFINITION: Two statements are logically equivalent if their truth values are thesame for all possible true/false combinations of their components.

(�Q → �P)(�P → �Q)

(Q → P)

P → Q

(�P � �Q)(P � Q)

P → Q

P → Q

EXAMPLE 5

Use a truth table to show that the statement and its contrapositiveare logically equivalent.

SolutionIn the following truth table, the final columns show that the implication and itscontrapositive have the same truth values for all combinations of P and Q.

�Q → �PP → Q

P Q

T T F F T TT F F T F FF T T F T TF F T T T T

�Q → �PP → Q�Q�P

P Q

T T TT F FF T TF F T

P → Q

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DEMORGAN’S LAWSIn the study of logic, DeMorgan’s Laws are used to describe the negations of the con-junction and disjunction. Augustin DeMorgan was a nineteenth-century Englishmathematician and logician.

THE TAUTOLOGYA truth table can be used to show that some statements are always true. For instance,

is always true.P � �P

A4 Logic Appendix

DEMORGAN’S LAWS1.

The negation of a conjunction is the disjunction of negations.2.

The negation of a disjunction is the conjunction of negations.[�(P � Q)] ↔ [�P � �Q]

[�(P � Q)] ↔ [�P � �Q]

EXAMPLE 7

Use DeMorgan’s Laws to write the negation of:

a) and 13 is prime. b) Clint is cool or Tim is handsome.

Solutiona) or 13 is composite (opposite of prime).b) Clint is not cool and Tim is not handsome.

2 � 3 � 5

2 � 3 � 5

EXAMPLE 8

Use a truth table to establish DeMorgan’s first law, .

SolutionWe need to show that and have identical truth values. The results are shown in the fourth and seventh columns.

NOTE: The final column is completed by looking at the columns headed �P and �Q.

[�P � �Q][�(P � Q)]

[�(P � Q)] ↔ [�P � �Q]

P Q

T T T F F F FT F F T F T TF T F T T F TF F F T T T T

�P � �Q�Q�P�(P � Q )P � Q

EXAMPLE 6

On the basis of Example 5, write a statement that is logically equivalent to “If aperson lives in London, then the person lives in England.”

SolutionThe given statement is true. Its contrapositive (also true) is “If a person does not live in England, then the person does not live in London.”

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Section 1 Truth Tables A5

In the following example, there are three component statements: P, Q, and R. Inorder to consider all true/false possibilities of these statements, we need eight hori-zontal rows in the table.

In Section 2 of this Logic Appendix, we will show that certain forms of deductivereasoning lead to conclusions that cannot be refuted. Such arguments are known asvalid arguments and are used throughout the study of mathematics. To establish thatan argument is valid, we must show that its premises and conclusion form a com-pound statement that is a tautology.

DEFINITION: A tautology is a statement that is true for all possible truth values of itscomponents.

EXAMPLE 9

Show that the statement is a tautology.

SolutionHere we look at the truth value of and then back to the column headed Qto determine the truth or falsity of

.

NOTE: A tautology must have a finalcolumn consisting only of T’s (true).

(P � Q) → Q

P � Q

(P � Q) → Q

P Q

T T T TT F F TF T F TF F F T

(P � Q ) → QP � Q

EXAMPLE 10

Is the statement a tautology?

SolutionTo list all possible truth value combinations for P, Q, and R, we use the eight hor-izontal rows shown. It will also be convenient to have one column headed .We will then decide on the truth/falsity of by considering thecolumns headed P and .

The statement is not a tautology because the final column con-tains an F.

P � (Q � R)

(Q � R)P � (Q � R)

Q � R

P � (Q � R)

P Q R

T T T T TT T F T TT F T T TT F F F TF T T T TF T F T TF F T T TF F F F F

P � (Q � R)Q � R

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A6 Logic Appendix

In Exercises 1 to 8, statement P is true, Q is true, and R isfalse. Classify each statement as true or false.

1. 2.

3. 4.

5. 6.

7. 8.

In Exercises 9 to 12, let “Mary is an accountant” and“Hamburgers are health food.” Write each symbolic

statement in words.

9. �Q

10.

11.

12.

In Exercises 13 to 18, form a truth table and determine allpossible truth values for the given statement. Is the givenstatement a tautology?

13.

14.

15.

16.

17.

18.

In Exercises 19 to 24, use DeMorgan’s Laws to write thenegation of the given statement.

19.

20.

21. Mary is an accountant or hamburgers are healthfood.

22. Mary is an accountant and hamburgers are healthfood.

23. It is cold and snowing.

24. We will go to dinner or to the movie.

25. Use a truth table to prove DeMorgan’s second law,.

(H I N T: See Example 8.)

26. Use a truth table to prove .(N OT E : This proof establishes that the converseand inverse of an implication are logicallyequivalent.)

27. Use a truth table to show that is a tautology.

28. Use a truth table to show that andare logically equivalent.

29. Use a truth table to show that is the nega-tion of .(H I N T: The truth values of these statements mustbe opposites.)

In Exercises 30 to 33, use the result of Exercise 29 to writethe negation of the given statement.

30. If it is medicine, then it tastes bad.

31. If I am good, then I can go to the movie.

32. If I am 18 or older, then I can vote.

33. If I study hard and make an A, then I can be a mem-ber of Phi Theta Kappa.

P → Q[P � �Q]

[(P � Q) � (P � R)][P � (Q � R)]

→ (P → R)[(P → Q) � (Q → R)]

[Q → P] ↔ [�P → �Q]

[�(P � Q)] ↔ [�P � �Q]

P � R

P � Q

[(P → Q) � P] → Q

[(P → Q) � Q] → P

(P � Q) → Q

(P � Q) → P

P � �P

P � �P

P � �Q

P → Q

P � Q

Q �P �

(P � Q) � RP � (Q � R)

P → QP → R

Q � RP � Q

P � �R�Q � R

Section 2 Valid ArgumentsWhat is an argument? By definition, an argument is a set of statements calledpremises, followed by a statement called the conclusion. In a valid argument, thetruth of the premises forces a conclusion that must also be true.

LAW OF DETACHMENTOne form of valid argument used often in this textbook is called the Law ofDetachment. This type of deductive reasoning takes the following form:

Section 1 Exercises

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Section 2 Valid Arguments A7

To prove that the Law of Detachment is valid, we need to establish thatis a tautology. In the truth table, study the five columns from left

to right for each possible true/false combination of P and Q. Consider the first row ofthe table:

If P and Q are true, then is true.

Because is true and P is true, the conjunction must betrue.

If is true and Q is true, then is true.

The student should verify the entries in each row and column of the table.

Proof of the Law of Detachment

[(P → Q) � P] → Q[(P → Q) � P]

[(P → Q) � P]P → Q

P → Q

[(P → Q) � P] → Q

The symbolic form of an argument is (conjunction of premises) implies(conclusion).

LAW OF DETACHMENT1. Premise 12. P Premise 2C. Q Conclusion

P → Q

EXAMPLE 1

Use the Law of Detachment to determine the conclusion in the following argu-ment. Assume that premises 1 and 2 are true.

1. If a person lives in London, then he lives in England.2. Simon lives in London.C. ?

SolutionSimon lives in England.

P Q

T T T T TT F F F TF T T F TF F T F T

[(P → Q) � P] → Q(P → Q) � PP → Q

EXAMPLE 2

Use the Law of Detachment to find the conclusion of the geometry argument.

1. If a triangle is isosceles, then it has two congruent sides.2. �ABC is isosceles.C. ?

Solution�ABC has two congruent sides.

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AN INVALID ARGUMENTA common error in reasoning occurs when one asserts the conclusion. This form of argument looks similar to the Law of Detachment. The mistake lies in the fact that the second premise is statement Q (and not P). Because the first premise of thisargument states that P implies Q, the argument is invalid (not valid).

To establish that an argument is not valid, we must show that its symbolic formis not a tautology. Again the form of the argument is (conjunction of premises)implies (conclusion). For the invalid argument preceding Example 3, the symbolicform is . To understand why this argument is not valid, see the lastcolumn in the following table.

LAW OF NEGATIVE INFERENCEA second form of valid argument, the Law of Negative Inference, is shown next andthen illustrated in Examples 4 and 5. This form of deductive reasoning was used tocomplete an indirect proof in Chapter 2.

[(P → Q) � Q] → P

A8 Logic Appendix

INVALID ARGUMENT1. Premise 12. Q Premise 2C. P Conclusion

P → Q

EXAMPLE 3

If possible, draw a conclusion in the following argument.

1. If Morgan works a lot of hours this week, Morgan can put money into savings.

2. Morgan put $50 in savings this week.C. ?

SolutionNo conclusion! Morgan may have money for savings because of his grand-parents’ generosity. We cannot conclude that Morgan worked a lot.

P Q

T T T T TT F F F TF T T T FF F T F T

[(P → Q) � Q] → P(P → Q) � QP → Q

LAW OF NEGATIVE INFERENCE1. Premise 12. �Q Premise 2C. �P Conclusion

P → Q

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Section 2 Valid Arguments A9

To prove that the Law of Negative Inference is valid requires that we establishthat is a tautology. Again, the columns of the following truthtable were developed from left to right, with the rightmost column showing that thesymbolic form of the argument is a tautology.

Proof of the Law of Negative Inference

[(P → Q) � �Q] → �P

EXAMPLE 4

Give the symbolic form of the Law of Negative Inference.

Solution(Conjunction of premises) implies (conclusion), so the form is

[(P → Q) � �Q] → �P

EXAMPLE 5

Use the Law of Negative Inference to determine the conclusion in the followingargument. Assume that premises 1 and 2 are true.

1. If a person plays on a major league baseball team, then he earns a good salary.

2. Bill McAllen does not earn a good salary.C. ?

SolutionBill McAllen does not play on a major league baseball team.

P Q �P �Q

T T F F T F TT F F T F F TF T T F T F TF F T T T T T

[(P → Q) � �Q] → �P(P → Q) � �QP → Q

EXAMPLE 6

Use the Law of Negative Inference to find the conclusion in the geometryargument.

1. If a quadrilateral is a parallelogram, then its opposite sides are parallel.

2. The opposite sides of quadrilateral MNPQ are not parallel.C. ?

SolutionQuadrilateral MNPQ is not a parallelogram.

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LAW OF SYLLOGISMIn every direct proof in geometry, there is a “chain” of conclusions that depend on aform of argument called the Law of Syllogism. This form of argument builds uponthree or more simple statements. For simplicity, we will illustrate this principle oflogic with three statements P, Q, and R.

In the proof of the Law of Syllogism, the truth table requires eight rows becausethere are eight different true/false combinations for three simple statements P, Q, andR. The method for determining the number of rows needed in a truth table follows.

Proof of the Law of Syllogism

If a statement is composed of n simple statements, then there are 2n rows inthe truth table.

A10 Logic Appendix

LAW OF SYLLOGISM1. Premise 12. Premise 2C. ConclusionP → R

Q → RP → Q

EXAMPLE 7

Use the Law of Syllogism to find the conclusion in the argument.

1. If an integer is even, then it has a factor of 2.2. If a number has a factor of 2, then it can be divided exactly

by 2.C. ?

SolutionIf an integer is even, then it can be divided exactly by 2.

P Q R

T T T T T T T TT T F T F F F TT F T F T T F TT F F F T F F TF T T T T T T TF T F T F T F TF F T T T T T TF F F T T T T T

[(P → Q) � (Q → R)] → (P → R)(P → Q) � (Q → R)P → RQ → RP → Q

EXAMPLE 8

Use the Law of Syllogism to find the conclusion in the geometric argument.

1. If a triangle is isosceles, then it has two congruent sides.2. If a triangle has two congruent sides, then it has two congruent

angles.C. ?

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Section 2 Valid Arguments A11

Many forms of deductive reasoning are used to reach conclusions. In each form,the truth table corresponding to the form of argument reveals a tautology.

In Exercises 1 to 4, use the Law of Detachment to draw aconclusion.

1. If two angles are complementary, the sum of theirmeasures is 90°. and are complementary.

2. If it gets hot this morning, we will have to turn onthe air conditioner. It is hot this morning.

3. If Tina goes ice skating, she will have a good time.Tina goes ice skating.

4. If Gloria is scheduled to perform at the concert, thenwe will go to the concert. Gloria is scheduled to per-form at the concert.

In Exercises 5 to 8, use the Law of Negative Inference todraw a conclusion.

5. If two angles are complementary, the sum of theirmeasures is 90°. m m .

6. If an animal lives in the zoo, it should have a com-panion of the same species. Fido does not have acompanion of the same species.

7. If Tom doesn’t finish the job, then I will not pay him.I did pay Tom for the job.

8. If the traffic light changes, then you can travelthrough the intersection. You cannot travel throughthe intersection.

In Exercises 9 to 12, use the Law of Syllogism to draw aconclusion.

9. If Izzi lives in Chicago, then she lives in Illinois. If aperson lives in Illinois, then she lives in the Midwest.

10. If you pay your tuition, then you will need to payadditional fees. If you need to pay additional fees,then you will need to write a check.

11. If Ken Travis gets a hit, then my favorite baseballteam will win the game. If my favorite baseball teamwins the game, then I will be happy.

12. If Tom speaks at the rally, the union members willlisten. If the union members listen, the union willdecide not to strike.

In Exercises 13 to 16, determine which arguments are valid.

13. 1. If I go to the football game, I’ll cheer for theCowboys.

2. I went to the football game.C. I cheered for the Cowboys.

14. 1. If I go to the football game, I’ll cheer for theCowboys.

2. I cheered for the Cowboys.C. I went to the football game.

15. 1. If Bill and Mary stop to visit, I’ll prepare a meal.2. Bill stopped to visit at 5 P.M.C. I prepared a meal.

16. 1. If it turns cold and snows, I’ll build a fire in thefireplace.

2. The temperature fell below freezing around 3 P.M.3. It began snowing before 5 P.M.C. I built a fire in the fireplace.

In Exercises 17 to 20, which law of reasoning was used toreach the conclusion?

17. 1. If it is cloudy, then it will rain.2. If it rains, the garden will grow.C. If it is cloudy, the garden will grow.

18. 1. If you leave the apartment unlocked, someonewill steal your CD player.

2. Your CD player was not stolen.C. You did not leave the apartment unlocked.

19. 1. If it snows more than 3 in. today, then we will goskiing.

2. It snowed 6 in. today.C. We will go skiing.

�2 � 90°�1 �

�2�1

SolutionIf a triangle is isosceles, then it has two congruent angles.

Section 2 Exercises

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20. 1. If the mosquitoes are bad, I won’t get out of the car.2. If I don’t get out of the car, we cannot have a

picnic.C. If the mosquitoes are bad, we cannot have a

picnic.

21. A form of deductive reasoning known as the Law ofDenial (or Denial of Alternative) is shown below.

a) Write the symbolic form of this argument.b) Complete a truth table to establish the validity of

this argument.

In Exercises 22 to 24, use the Law of Denial (see Exercise21) to draw a conclusion.

22. Terry is sick or hurt.

Terry is not hurt.

23. Mary’s family will visit us at Thanksgiving or atChristmas.

Mary’s family did not visit us at Thanksgiving.

24. Wendell will have to study geometry or he will failthat course.

Wendell did not fail the geometry course.

In Exercises 25 to 27, complete a truth table to validateeach form of reasoning. Do not merely copy the truthtables in this section.

25. Law of Detachment

26. Law of Negative Inference

27. Law of Syllogism

A12 Logic Appendix

LAW OF DENIAL1.2. �QC. P

P � Q

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