Logic Appendix T F F T Section 1 Truth Tables Recall that a statement is a group of words or symbols that can be classified collectively as true or false. The claim 5 7 12 is a true statement, whereas A pentagon has six sides is a false statement. However, Man overboard! is not a statement at all. Symbolically, we represent statements by letters such as, Q, and R. The negation of a given statement is written ; when is true, is false, and vice versa. This relationship between a statement and its negation can be expressed by a truth table. The first line of the truth table shown indicates If is true, then is false. The second line in the table indicates When is false, is true. Suppose that is the true statement Abraham Lincoln lived in Illinois. Then Abraham Lincoln did not live in Illinois is false. EXAMLE 1 Determine which of the following are statements. For each one that is a statement, is it true or false? a) 4 3 5 c) Are you Mike? b) Babe Ruth played baseball. d), if is true. a) False statement c) Not a statement b) True statement d) False statement A truth table is a valuable tool for examining the truth of a statement that is more complex. What exactly is a truth table? DEFINITION: A truth table is a table that provides the truth values of a statement by considering all possible true/false combinations of the statement s components. Q Q T T T T F F F T F F F F CONJUNCTION Statements can be combined to form compound statements. For example, a statement of the form and Q is called the conjunction of and Q. In symbols, the conjunction is written Q. For the conjunction to be true, it is necessary for to be true and for Q to be true. If either statement is false, the conjunction is false. To allow for all possible true/false combinations, four rows are needed in the truth table of the conjunction. When is true, Q may be true or false (two rows of the table). When is false, Q may be true or false (two additional rows). A1
A2 Logic Appendix EXAMLE 2 Let Babe Ruth played baseball and Q 4 3 5. Classify as true or false: a) Q b) Q a) The conjunction is false because Q is false. b) The conjunction is true because is true and Q is also true. Q Q T T T T F T F T T F F F DISJUNCTION A compound statement of the form or Q is called the disjunction of and Q. In symbols, the disjunction is written Q. For the disjunction to be true, either is true, Q is true, or both and Q are true. A disjunction is false only if and Q are both false. To understand the first line in the truth table, consider this situation: You can join the Math Club if you have an A average or you are enrolled in a mathematics class. If you satisfy both requirements, you may still join the club. EXAMLE 3 Let Babe Ruth played baseball and Q 4 3 5. Classify as true or false: a) Q b) Q a) The disjunction is true because is true even though Q is false. b) The disjunction is true because is true and Q is also true. In some cases, we use parentheses to clarify the meaning of a compound statement. In this context, parentheses are given priority just as they are in numerical expressions. See Example 4. EXAMLE 4 Where, Q, and R are statements, suppose that is true, Q is false, and R is false. Classify the statement ( Q R) as true or false. The statement in parentheses is a disjunction of the form T or F and is true. Then ( Q R) is a conjunction of the form T and T, so the given statement is true. NOTE: The steps used to determine the truth of the statement in Example 4 can be written as follows: ( Q R) T (T F) T (T) T IMLICATION The final compound statement that we consider is of the form If, then Q. This statement is called an implication or a conditional statement. In symbols, we write
Section 1 Truth Tables A3 Q T T T T F F F T T F F T. The conditional statement makes a promise and fails to satisfy the conditions of this promise only when is true and Q is false (see the truth table). Consider the claim If you are good, then I ll give you a dollar. The only way the claim is false is when you are good, but I don t give you the dollar. In, 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 the same. 2. The negation of a compound statement has a particular form. For instance, we can show that the negation of ( Q) is ( Q). 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, then Q is expressed symbolically by. Related to the conditional are its Converse If Q, then. (Q ) Inverse If not, then not Q. ( Q) Contrapositive If not Q, then not. ( Q ) LOGICAL EQUIVALENCE OF STATEMENTS DEFINITION: Two statements are logically equivalent if their truth values are the same for all possible true/false combinations of their components. In Example 5, we will show that an implication and its contrapositive are logically equivalent. Because the example involves two simple statements and Q, the truth table shown has four horizontal rows. At the top of the six vertical columns of the truth table are and Q as well as certain statements that involve statement, statement Q, or both. Because our goal is to compare the truth values of and Q, the column headings of the truth table are chosen accordingly. EXAMLE 5 Use a truth table to show that the statement and its contrapositive Q are logically equivalent. In the following truth table, the final columns show that the implication and its contrapositive have the same truth values for all combinations of and Q. Q Q T T F F T T T F F T F F F T T F T T F F T T T T Q A double arrow is used to show that two statements are logically equivalent. In Example 5,. () ( Q )
A4 Logic Appendix EXAMLE 6 On the basis of Example 5, write a statement that is logically equivalent to If a person lives in London, then the person lives in England. The 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. DEMORGAN S LAWS In the study of logic, DeMorgan s Laws are used to describe the negations of the conjunction and disjunction. Augustin DeMorgan was a nineteenth-century English mathematician and logician. 1. 2. DEMORGAN S LAWS [ ( Q)] [ Q] The negation of a conjunction is the disjunction of negations. [ ( Q)] [ Q] The negation of a disjunction is the conjunction of negations. EXAMLE 7 Use DeMorgan s Laws to write the negation of: a) 2 3 5 and 13 is prime. b) Clint is cool or Tim is handsome. a) 2 3 5 or 13 is composite (opposite of prime). b) Clint is not cool and Tim is not handsome. EXAMLE 8 Use a truth table to establish DeMorgan s first law, [ ( Q)] [ Q]. We need to show that [ ( Q)] and [ Q] have identical truth values. The results are shown in the fourth and seventh columns. Q Q ( Q) Q Q T T T F F F F T F F T F T T F T F T T F T F F F T T T T NOTE: The final column is completed by looking at the columns headed and Q. THE TAUTOLOGY A truth table can be used to show that some statements are always true. For instance, is always true.
Section 1 Truth Tables A5 DEFINITION: A tautology is a statement that is true for all possible truth values of its components. EXAMLE 9 Show that the statement ( Q) Q is a tautology. Here we look at the truth value of Q and then back to the column headed Q to determine the truth or falsity of ( Q) Q. Q NOTE: A tautology must have a final column consisting only of T s (true). Q ( Q) Q T T T T T F F T F T F T F F F T In the following example, there are three component statements:, Q, and R. In order to consider all true/false possibilities of these statements, we need eight horizontal rows in the table. EXAMLE 10 Is the statement (Q R) a tautology? To list all possible truth value combinations for, Q, and R, we use the eight horizontal rows shown. It will also be convenient to have one column headed Q R. We will then decide on the truth/falsity of (Q R) by considering the columns headed and (Q R). Q R Q R (Q R) T T T T T T T F T T T F T T T T F F F T F T T T T F T F T T F F T T T F F F F F The statement (Q R) is not a tautology because the final column contains an F. In Section 2 of this Logic Appendix, we will show that certain forms of deductive reasoning lead to conclusions that cannot be refuted. Such arguments are known as valid arguments and are used throughout the study of mathematics. To establish that an argument is valid, we must show that its premises and conclusion form a compound statement that is a tautology.
A6 Logic Appendix In Exercises 1 to 8, statement is true, Q is true, and R is false. Classify each statement as true or false. In Exercises 9 to 12, let Mary is an accountant and Q 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 all possible truth values for the given statement. Is the given statement a tautology? 13. 14. 15. 16. 17. 18. In Exercises 19 to 24, use DeMorgan s Laws to write the negation of the given statement. 19. Q Q ( Q) ( Q) Q Q Section 1 Exercises 1. Q R 2. R 3. Q 4. Q R 5. R 6. 7. (Q R) 8. ( Q) R [() Q] [() ] Q 21. Mary is an accountant or hamburgers are health food. 22. Mary is an accountant and hamburgers are health food. 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, [ ( Q)] [ Q]. (HINT: See Example 8.) 26. Use a truth table to prove [Q ] [ Q]. (NOTE: This proof establishes that the converse and inverse of an implication are logically equivalent.) 27. Use a truth table to show that [() (Q R)] ( R) is a tautology. 28. Use a truth table to show that [ (Q R)] and [( Q) ( R)] are logically equivalent. 29. Use a truth table to show that [ Q] is the negation of. (HINT: The truth values of these statements must be opposites.) In Exercises 30 to 33, use the result of Exercise 29 to write the 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 member of hi Theta Kappa. 20. R Section 2 Valid Arguments What is an argument? By definition, an argument is a set of statements called premises, followed by a statement called the conclusion. In a valid argument, the truth of the premises forces a conclusion that must also be true. LAW OF DETACHMENT One form of valid argument used often in this textbook is called the Law of Detachment. This type of deductive reasoning takes the following form:
Section 2 Valid Arguments A7 LAW OF DETACHMENT 1. remise 1 2. remise 2 C. Q Conclusion EXAMLE 1 Use the Law of Detachment to determine the conclusion in the following argument. 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. Simon lives in England. The symbolic form of an argument is (conjunction of premises) implies (conclusion). To prove that the Law of Detachment is valid, we need to establish that [() ] Q is a tautology. In the truth table, study the five columns from left to right for each possible true/false combination of and Q. Consider the first row of the table: If and Q are true, then is true. Because is true and is true, the conjunction [() ] must be true. If [() ] is true and Q is true, then [() ] Q is true. The student should verify the entries in each row and column of the table. roof of the Law of Detachment Q () [() ] Q T T T T T T F F F T F T T F T F F T F T EXAMLE 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. ABC has two congruent sides.
A8 Logic Appendix AN INVALID ARGUMENT A 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 ). Because the first premise of this argument states that implies Q, the argument is invalid (not valid). INVALID ARGUMENT 1. remise 1 2. Q remise 2 C. Conclusion EXAMLE 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. No conclusion! Morgan may have money for savings because of his grandparents generosity. We cannot conclude that Morgan worked a lot. To establish that an argument is not valid, we must show that its symbolic form is not a tautology. Again the form of the argument is (conjunction of premises) implies (conclusion). For the invalid argument preceding Example 3, the symbolic form is [() Q]. To understand why this argument is not valid, see the last column in the following table. Q () Q [() Q] T T T T T T F F F T F T T T F F F T F T LAW OF NEGATIVE INFERENCE A second form of valid argument, the Law of Negative Inference, is shown next and then illustrated in Examples 4 and 5. This form of deductive reasoning was used to complete an indirect proof in Chapter 2. LAW OF NEGATIVE INFERENCE 1. remise 1 2. Q remise 2 C. Conclusion
Section 2 Valid Arguments A9 EXAMLE 4 Give the symbolic form of the Law of Negative Inference. (Conjunction of premises) implies (conclusion), so the form is [() Q] EXAMLE 5 Use the Law of Negative Inference to determine the conclusion in the following argument. 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. Bill McAllen does not play on a major league baseball team. To prove that the Law of Negative Inference is valid requires that we establish that [() Q] is a tautology. Again, the columns of the following truth table were developed from left to right, with the rightmost column showing that the symbolic form of the argument is a tautology. roof of the Law of Negative Inference Q Q () Q [() Q] T T F F T F T T F F T F F T F T T F T F T F F T T T T T EXAMLE 6 Use the Law of Negative Inference to find the conclusion in the geometry argument. 1. If a quadrilateral is a parallelogram, then its opposite sides are parallel. 2. The opposite sides of quadrilateral MNQ are not parallel. Quadrilateral MNQ is not a parallelogram.
A10 Logic Appendix LAW OF SYLLOGISM In every direct proof in geometry, there is a chain of conclusions that depend on a form of argument called the Law of Syllogism. This form of argument builds upon three or more simple statements. For simplicity, we will illustrate this principle of logic with three statements, Q, and R. LAW OF SYLLOGISM 1. remise 1 2. Q R remise 2 C. R Conclusion EXAMLE 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. If an integer is even, then it can be divided exactly by 2. In the proof of the Law of Syllogism, the truth table requires eight rows because there are eight different true/false combinations for three simple statements, Q, and R. The method for determining the number of rows needed in a truth table follows. If a statement is composed of n simple statements, then there are 2 n rows in the truth table. roof of the Law of Syllogism Q R Q R R () (Q R) [() (Q R)] ( 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 EXAMLE 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.
Section 2 Valid Arguments A11 If a triangle is isosceles, then it has two congruent angles. 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. Section 2 Exercises In Exercises 1 to 4, use the Law of Detachment to draw a conclusion. 1. If two angles are complementary, the sum of their measures is 90. 1 and 2 are complementary. 2. If it gets hot this morning, we will have to turn on the 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, then we will go to the concert. Gloria is scheduled to perform at the concert. In Exercises 5 to 8, use the Law of Negative Inference to draw a conclusion. 5. If two angles are complementary, the sum of their measures is 90. m 1 m 2 90. 6. If an animal lives in the zoo, it should have a companion of the same species. Fido does not have a companion 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 travel through the intersection. You cannot travel through the intersection. In Exercises 9 to 12, use the Law of Syllogism to draw a conclusion. 9. If Izzi lives in Chicago, then she lives in Illinois. If a person lives in Illinois, then she lives in the Midwest. 10. If you pay your tuition, then you will need to pay additional 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 baseball team will win the game. If my favorite baseball team wins the game, then I will be happy. 12. If Tom speaks at the rally, the union members will listen. If the union members listen, the union will decide 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 the Cowboys. 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 the Cowboys. 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.M. C. I prepared a meal. 16. 1. If it turns cold and snows, I ll build a fire in the fireplace. 2. The temperature fell below freezing around 3.M. 3. It began snowing before 5.M. C. I built a fire in the fireplace. In Exercises 17 to 20, which law of reasoning was used to reach 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, someone will 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 go skiing. 2. It snowed 6 in. today. C. We will go skiing.
A12 Logic Appendix 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 of Denial (or Denial of Alternative) is shown below. LAW OF DENIAL 1. Q 2. Q C. a) Write the symbolic form of this argument. b) Complete a truth table to establish the validity of this argument. 23. Mary s family will visit us at Thanksgiving or at Christmas. Mary s family did not visit us at Thanksgiving. 24. Wendell will have to study geometry or he will fail that course. Wendell did not fail the geometry course. In Exercises 25 to 27, complete a truth table to validate each form of reasoning. Do not merely copy the truth tables in this section. 25. Law of Detachment 26. Law of Negative Inference 27. Law of Syllogism In Exercises 22 to 24, use the Law of Denial (see Exercise 21) to draw a conclusion. 22. Terry is sick or hurt. Terry is not hurt.