CHAPTER ONE STATEMENTS, CONNECTIVES AND EQUIVALENCES A unifying concept in mathematics is the validity of an argument To determine if an argument is valid we must examine its component parts, that is, the sentences or statements that comprise the argument. Such sentences are called True/False (TF) statements. In this chapter we will examine individual TF statements and the relationship between combinations of TF statements. Objectives After completing Chapter One, the student should be able to: Identify True/False statements. Translate an English sentence into a symbolic logic statement Translate a symbolic logic statement into an English sentence. Determine the truth value of a compound statement. Determine the truth value of a variable given the truth value of a compound statement. Construct truth tables for compound statements Recognize logically equivalent statements. Write a statement that is logically equivalent to a given statement Construct the converse, inverse, and contrapositive of a conditional statement. Write the negation for each logical connective 1.1 Introduction to Logic TF Statements A TF statement is a declarative sentence to which we can assign a truth value of either true or false, but not both at the same time. Example I "Groundhog Day is celebrated on February 2 nd " is a TF statement since it is a declaration, and is either true or false. Example 2 "Bill Clinton was the first president of the United States to be impeached" is a TF statement since, as in the previous example, it is a declarative sentence. In this case the TF statement is false since Andrew Johnson, the 17 th president, was impeached. Example 3 "Are you going to climb Mt. Everest?" is not a TF statement since it is a question, not a declarative sentence. Example 4 "Put your cell phone away!" is not a TF statement. It is a command, not a declarative sentence. 1
Representing Statements using Variables TF statements are often represented by variables. In this class the variables used will be lowercase letters. p: Popcorn is a popular snack at the movies. would be read 'p represents (or stands for) the statement 'Popcorn is a popular snack at the movies.' " The choice of the letter p to symbolize this sentence is convenient, but any letter could have been used. Negations To negate a TFstatement is to change its truth value. Any TF statement may be negated by using the word not. It may also be negated by placing the words it is not true that in front of the statement. The negation is denoted by the symbol. Thus, the statement ~p is read "not p." The "~" symbol is also used to mean it is not true that. Example 5 The TF statement "Pluto is not classified as a planet" can be symbolized as ~p. Notice that this is really the negation of the TF statement "Pluto is classified as a planet." Example 6 Let q: The quagga is an extinct relative of the plains zebra of South Africa. Then ~q: It is not true that the quagga is an extinct relative of the plains zebra of South Africa, or simply "The quagga is not an extinct relative of the plains zebra of South Africa." Example 7 Let c: Carbonatites are igneous rocks. Then ~c: "It is not true that carbonatites are igneous rocks" or simply "Carbonatites are not igneous rocks" Connectives A connective is a word or group of words used to join two TF statements In our study of logic, the four basic connectives we will use are the conjunction, the disjunction, the conditional, and the biconditional. Simple and Compound Statements If a TF statement does not contain a connective, it is called a simple statement. If two or more simple statements are joined by one or more connectives, the resulting statement is called a compound statement. Conjunctions If we join two TF statements by the word and, we call the resulting compound statement the conjunction of the two TF statements. The symbol for the conjunction is. The statement p q is read 'p and q" or 'p conjunction q." In logic, the word "but" also means "and." Example 8 Let v: The voting age in the US is 18. Let d: The driving age in the US varies from state to state. The conjunction "The voting age in the US is 18 and the driving age in the US varies from state to state" is written symbolically as: v d. > > > 2
Example 9 Let m: The Acela Express is a fast moving train. Let a: The Acela Express is a costly train. The symbolization m A a means: > "The Acela Express is a fast moving train and it is a costly train" or alternatively, "The Acela Express is a fast moving but costly train." Example 10 Let d: I like to buy coffee from Dunkin' Donuts. Let s: I like to buy coffee from Starbucks. Then d A ~s represents the compound statement: > "I like to buy coffee from Dunkin' Donuts and I do not like to buy coffee from Starbucks," or simply "I like to buy coffee from Dunkin' Donuts but not from Starbucks." If a conjunction contains two negated statements, as in ~p the statement as "neither p nor q." > ~q, we sometimes read Example 11 Let s: Shoehorns are shoes. Let h: Shoehorns are horns. The statement "Shoehorns are not shoes and shoehorns are not horns" can be written symbolically as ~s ~h. Alternatively, we can say that "Shoehorns are neither shoes nor horns." > Example 12 Let g: The guinea pig is from Guinea. Let p: The guinea pig is a pig. The symbolization of the compound statement "The guinea pig is neither from Guinea nor is it a pig" is: ~g ~p Example 13 Using the underlined letters, the statement "A pocketbook is neither a pocket nor a book" can be symbolized as ~k ~b > > Disjunctions If we join two TF statements by the word or, the resulting compound statement is called the disjunction of the two statements. The symbol for the disjunction is The disjunctive statement p q is read "p or q" or 'p disjunction q." Sometimes the word either is implied as the correlative of the word or. > Example 14 Letp: I will pass physics. Lets: I will go to summer school. Then the disjunction p v s represents the compound statement "Either I will pass physics or I will go to summer school" or simply "I will pass physics or go to summer school." > 3
Example 15 Let c: In NY State, bridge tolls are paid in cash. Let e: In NY State, bridge tolls are paid with E ZPass. Then the disjunction c V e represents the compound statement "In NY State, bridge tolls are paid either in cash or with E ZPass." Conditionals Given two statements and q, a compound statement of the form "if p then q" is called a conditional statement or an implication. The symbol for a conditional statement is The conditional statement p q is read "p implies q". or "if p then q." The left hand side (LHS) of the conditional statement,p, is called its antecedent. The right hand side (RHS) of the conditional statement, q, is called its consequent. Example 16 Letj: I have a part time job. Let p: I pay for my auto insurance. The implication "If I have a part time job then I pay for my auto insurance" is symbolically written as: j p, while p j is read "If I pay for my auto insurance then I have apart time job." As we shall see, these two statements are not Example 17 Let r: I order spareribs. Let C: I go to a Chinese restaurant. Consider the statement "I order spare ribs if I go to a Chinese restaurant" This statement may be rephrased as "If I go to a Chinese restaurant, then I order spare ribs." Notice that the antecedent of the statement is c, and the consequent is r. Thus the statement is represented symbolically as c r. Example 18 Using the underlined letters, the symbolization of "The food will spoil if left unrefrigerated" is ~u s. Example 19 Using the underlined letters, the symbolization for "If credit card bills are not paid on time, an interest charge will be added to the bill" is ~p a. Biconditionals A biconditional statement is formed when two statements, p and q, are joined by the words "if and only if." The symbol for the biconditional statement is. Thus, the biconditional p q is read 'p if and only if q." Notice that the symbol for the biconditional has a double headed arrow. This really means that p q is the same as p q and q p. That is, p q is the same as(p q) (q p). V 4
Example 20 Let m: The mountain gorilla will survive. Let p: Poaching is halted. Then m p is read "The mountain gorilla will survive if and only if poaching is halted." Example 21 Using the underlined letters, the symbolization for "MRI's are prescribed if and only if there are no other options" is m o The Role of Parentheses in Compound Statements Consider the statement "Emeril will buy apples or bananas and cherries." This statement has three simple ideas: a: Emeril will buy apples, b: Emeril will buy bananas and C: Emeril will buy cherries. The statement contains an "or" as well as an "and" connective. Therefore, a reasonable question to ask is, "Does it matter whether we think of this statement as the disjunction a V (b c), or as the conjunction (a V b) C?" The following table summarizes the possible ways Emeril could fill a shopping cart, depending on where the parentheses are placed. < < These results are not the same. This is because there is ambiguity in the English statement as it was written. However, a mathematical statement must not have any ambiguity. It must have one and only one meaning. In an English sentence, the ambiguity is often remedied by using a comma to separate ideas In order to construct compound statements that are non ambiguous in logic, parentheses are often required to convey the intended meaning. Be aware that if a compound sentence contains two or more connective symbols, parentheses must be used to convey the true meaning of the statement. 5
. Example 22 Let C: i like carrots. Letp: I like peas. Explain the difference between ~(c p) and ~c p. Solution The statement ~(c p) is translated as "It is not true that I like both carrots and peas." The statement ~c p means "I don't like carrots but I do like peas." > > >> Example 23 Is the following statement a conjunction or a disjunction? "I will order the soup and the salad, or the club steak" Solution Noticing the placement of the comma, we see that the statement is a disjunction The symbolization is (s d) V c > Example 24 Is the following statement a conditional or a conjunction? "I will go to the night club, and if you are not there I will leave" Solution Noticing the placement of the comma, we see that the statement is a conjunction. The symbolization is n (~y l ). > Example 25 Is the following statement a conditional or a conjunction? "If the team comes in last place, then they will fire their manager and make a trade for a better second baseman." Solution This is a conditional statement. The symbolization is 1 (m s). > In Class Exercises and Problems for Section 1.1 In Exercises 1 15, use the underlined letters to write the sentences in symbolic form: 1. Soccer and basketball are international sports. 2. If it is 3 pm in New York, then it is noon in San Francisco. 3 Either you pay your taxes on time or you will incur a penalty. 4. You can vote in a federal election if and only if you are at least 21 years of age. 5. Okapis, giraffe like animals, are not found in Canada. 6. An angle is formed if two rays meet at a common endpoint. 7. Bald eagles are neither bald nor are they found in South America. 8. It is recommended that you take calcium supplements and lift weights if you have been diagnosed with osteoporosis. 9. If a team loses the Super Bowl, then there will be neither a parade nor a championship ring. 10. Solving a sudoku puzzle is time consuming but enjoyable. 6
11.If she has a bronze medal she did not win, but if she has a silver medal she came in second. 12. If the Senate and House pass the bill, then it will become a law and the police will enforce it. 13. The roads will become icy if the rain changes to sleet. 14. If you go to the beach and do not apply sunscreen you will get a sunburn. 15. The rich dark Turkish coffee was poured into the large green ceramic mug.(choose your own symbols) In Exercises 16 18, let p: She works part time. Let q: She owns a car. Let r: She goes to college. Let s: She lives at home. Express each of the following compound statements as an English sentence. In Problems 19 41, use the underlined letters to express the statement symbolically. 7
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1.2 Connectives and their Truth Tables A truth table indicates the instances when a TF statement is true and when it is false. Truth values are displayed within the table using the capital letters T and F. Negations The following truth table shows us that the negation of a true statement is false, while the negation of a false statement is true. p ~p T F Conjunctions Let us examine the truth values of a conjunction. By the nature of the word "and," if a conjunctive statement is to be true, both of its component sentences must be true. If either sentence is false, the conjunction is false. Thus, the only time the conjunction >p q is true is when p and are both true. F T Example 1 A prestigious organization at Rafael's college is known as the Gold Club. Requirements for membership are listed in the club's charter, and read: "The student must have a GPA of at least 3.6 and must be an officer of a campus club." Rafael wishes to apply for membership in this club. Let p: Rafael's GPA is at least 3.6. Let q: Rafael is an officer of a campus club. Determine the truth values of p and q such that Rafael will be eligible for membership in the Gold Club. 9
Disjunctions In the English language, the word "or" is often used in the exclusive sense. When we say "Either we will go to the movies or go out to dinner," we often mean that we will do one of these two activities, but not both. In logic, the "or" is inclusive. In the above example, we mean that we will go to the movies, or go to dinner, or do both. The following truth table summarizes our results about disjunctions. Notice that the disjunction p V q is false only when p and are both false. Another, organization at the college in Example 1 is known as the Silver Club. Requirements for membership are listed in the Silver Club's charter, and read: "The student must have a GPA of at least 3.6 or must be a member of a sports team on campus." Suzie wishes to apply for membership in this club. Let p: Suzie's GPA is at least 3.6. Let q: Suzie is a member of a sports team on campus. Determine the truth values of p and q such that Suzie will be accepted for membership in the Silver Club. Conditionals The truth table for a conditional statement is less intuitive than the truth table for a conjunction or for a disjunction. Mathematicians have agreed that the only time the conditional statement p q is false is when the LHS, p, is true and the RHS, q, is false. The following truth table shows the truth values for a conditional statement. 10
Example 3 Ada has not been feeling well. She asks the advice of her doctor who prescribes a certain medication. The doctor promises Ada that if she takes the medication, she will feel better. Let p: Ada takes the medication and let q: Ada feels better. Therefore, the doctor's promise is represented symbolically as p q Examine all four sets of truth values for p and q, and in each case, decide if the doctor told Ada the truth. Solution As with the previous two connectives, let's examine the truth table for the conditional statement. Biconditionals The truth table for the biconditional is shown below. Notice that for a biconditional to be true, its LHS and the RHS must have the same truth value That is, for the biconditional statement p q to be true, both sides of the biconditional must be true, or both sides must be false. 11
Example 4 A meteorologist alerts the tn state area of a cold front headed our way. He warns that there will be a major snowfall if and only if the cold front brings subzero (Celsius) temperatures. Let p: There is a major snowfall. Let q: The cold front brings subzero temperatures. When is the meteorologist's forecast correct? In the first row we see that when p is true and q is true, the biconditional statement is true. In the second row twe see that when p is true and q is false, the biconditional statement is false. In the third row we see that when p is false and q is true, the biconditional statement is false. In the fourth row we see that when p is false and q is false, the biconditional statement is true. Truth Table Summary There are four sentences that stress the main points found in the truth tables for conjunction, disjunction, conditional and biconditional statements They are: 12
Example 5 Suppose p is a statement that we know is true and q is a statement that we know is false. Then True or False? In Class Exercises In Exercises 1 38, fill in each blank with either true, false, or can't be determined (CBD) due to insufficient information. 13
1.3 Evaluating TF Statements We have seen that the truth value of a compound statement depends on the truth values of its individual simple sentences and the particular connective used. Sometimes we wish to know the truth value when we combine more than two simple statements into a compound statement. The strategy we use to obtain the truth value is to consider the truth values two at a time. 14
Example 3 If p is true but q and rare false, find the truth value of (p q) ~ r. 15
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1.4 Using Truth Tables Introduction Often, it is desirable to know the truth value of a compound statement for every different combination of truth values of its component simple statements. We accomplish this with the use of truth tables. We have already seen this technique when we constructed truth tables for negations, conjunctions, disjunctions, implications, and biconditionals. T F F T T F 17
Tautologies Statements that are always true, regardless of the truth values of the simple statements involved are called tautologies. Compact Truth Tables Rather than constructing a column in a truth table for every TF statement in a complicated compound statement, and then a column for each sub statement, we often construct a more compact truth table that yields the same result. Solution We begin by noting that the truth table will have four rows since there are two simple sentences. T F Truth Tables with Three Simple Statements When only two variables are involved in the construction of a compound sentence, four different sets of truth values must be considered. This results in a truth table that has four rows. If a compound statement involves three simple statements, there will be exactly eight ways that the truth values for these variables can combine. This will result in a truth table that has eight rows. To insure that we account for all possible combinations of truth values without duplication, we enter the truth values for the three variables by column. The first column has four entries of T followed by four entries of F. The second column contains two entries of T followed by two entries of F. This pattern is then repeated. The last column has the truth values alternating, beginning with T. 18
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1.5 Equivalences Part I Introduction Two compound statements that always have the same truth value, regardless of the truth values of the variables involved, are said to be logically equivalent. When two statements are joined by the symbol, the resulting expression is referred to as an equivalence. Equivalences allow us to express the same thought in two different, but logically equivalent ways. Equivalences are used extensively in the remainder of this chapter and in many of the chapters that follow. While there are infinitely many equivalences, a few are used with such great frequency as to warrant special attention. The double negation equivalence: 21
Example 1 Let d: Fractions can be converted to decimals. Then ~d : Fractions cannot be converted to decimals. The statement "It is not true that fractions cannot be converted to decimals" symbolized as ~(~d), is logically equivalent to the statement "Fractions can be converted to decimals." The truth table verifies this relationship. Example 3 Let e: The Earth revolves around the sun. Let m: The moon revolves around the Earth. Use the commutative equivalence to write a statement equivalent to "The Earth revolves around the sun and the moon revolves around the Earth." The moon revolves around the Earth and the Earth revolves around the sun. m e > 22
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In Class Exercises For Exercises 1 11, complete each statement by applying the indicated equivalence. 25
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Example 5 Use the contrapositive equivalence to rewrite each of the following statements. a. r > w b. q m c. (aab) *' 'p 27
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