Module 4 - Logic Statements and Quantifiers PDF

MODULE 4 Intended Learning Outcomes: a. Validate the different propositions. Chapter 1: Logic The foundation of a sound decision is a good and reliable data gathering procedures. The credibility of the output relies much on how data is gathered and managed. Good data produces good result....

MODULE 4 Intended Learning Outcomes: a. Validate the different propositions. Chapter 1: Logic The foundation of a sound decision is a good and reliable data gathering procedures. The credibility of the output relies much on how data is gathered and managed. Good data produces good result. Lesson 1: Logic Statement and Quantifiers Logic Statements Every language contains different types of sentences, such as statements, questions, and commands. For example, QUESTION: “Is the test today?” COMMAND: “Go get the newspaper.” OPINION: “This is a nice car.” STATEMENT: “Sogod is a municipality of Southern Leyte.” The symbolic logic that Boole created applies only to sentences that are statements. 1  Example 1: Identify Statement Check Your Progress 1 Determine whether each sentence is a statement. a. Open the door. b. 7055 is a large number. c. In the year 2024, the president of the United States will be a woman. d. x>3 Connecting simple statements with words and phrases such as and, or, if …then, and if and only if creates a compound statement. George Boole used symbols such as p, q, r, and s to represent simple statements and the symbols , , , →, and  to represent connectives. 2 Logical Connectives A logical connective is the mathematical equivalent of a conjunction (a word or symbol that joins two sentences to produce a new one). Below are the logical connectives together with the symbols used: Truth Value and Truth Tables The truth value of a simple statement is either true (T) or false (F). The truth value of a compound statement depends on the truth values of its simple statements and its connectives. A truth table is a table that shows the truth value of a compound statement for all possible truth values of its simple statements. The negation of the statement “ Today is Friday.” is the statement “Today is not Friday.”in symbolic logic, the tilde symbol ~ is used to denote the negation of a statement. If a statement p is true, its negation ~𝑝 is false, and if a statement p is false, its negation ~𝑝 is true. The negation of the negation of a statement is the original statement. Thus ~(~𝑝 ) can be replaced by p in any statement. See table below. Truth Table for ~𝒑 p ~𝒑 T F F T Example 2: Write the negation of a statement Write the negation of each statement. a. Ellie Goulding is an opera singer. b. The dog does not need to be feed. 3 Solution a. Ellie Goulding is not an opera singer. b. The dog needs to be feed. Check Your Progress 2 Write the negation of each statement. a. The Queen Mary 2 is the world’s largest cruise shi. b. The fire engine is not real. Example 3: Write Compound Statement in Symbolic Form Consider the following simple statements. p: Today is Friday. q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. a. Today is Friday and it is raining. b. It is not raining and I am going to a movie. c. I am going to the basketball game or I am going to a movie. d. If it is raining, then I am not going to the basketball game. Solution: a. p q b. ~𝒒 r c. ~𝐬 r d. q → s Check Your Progress 3 Use p, q, r, and s as defined in Example 3 to write the following compound statements in symbolic form. a. Today is not Friday and I am going to a movie. b. I am going to the basketball game and I am not going to a movie. c. I am going to the movie if and only if it is raining. d. If today is Friday, then I am not going to a movie. Example 4: Translate Symbolic statements Consider the following statements. p: The game will be played in Atlanta. q: The game will be shown on CBS. r: The game will not be shown on ESPN. s: The Mets are favored to win. 4 Write each of the following symbolic statements in words. a. q p b. ~𝒓  s c. s  ~𝒑 Solution: a. The game will be shown on CBS and the game will be played in Atlanta. b. The game will be shown on ESPN and the Mets are favored to win. c. The Mets are favored to win if and only if the game will not be played in Atlanta. Check Your Progress 4 Consider the following statements. e: All men are created equal. r: I am trading places. a: I get Abe’s place. g: I get George’s place. Write each of the following symbolic statements in words. a. e  ~𝒕 b. a  ~𝒕 c. e → t d. t  g Compound Statements and Grouping Symbols If a compound statement is written in symbolic form, then parentheses are used to indicate which simple statements are grouped together. Symbolic form The parentheses indicate that: p  (q  ~𝒓 ) q and ~𝒓 are grouped together. (p  q )  r p and q are grouped together. (p  ~𝒒) → ( r  𝒔) p and ~𝒒 are grouped together. r and 𝒔 are grouped together. If a compound statement is written as an English sentence, then a comma is used to indicate which simple statements are grouped together. Statements on the same side of a comma are grouped together. English sentence The comma indicates that: p, and q or 𝐧𝐨𝐭 𝒓. q and ~𝒓 are grouped together because they are both on the same side of the comma. p and q , or r p and q are grouped together because they are both on the same side of the comma. If p and 𝐧𝐨𝐭 𝒒, p and ~𝒒 are grouped together because they are both to the left of 𝐭𝐡𝐞𝐧 r or 𝒔. the comma. r and 𝒔 are grouped together because they are both to the right of the comma. If a statement in symbolic form is written as an English sentence, then the simple statements that appear together in parentheses in the symbolic form will all be on the same side of the comma that appears in the English sentence. 5  Example 5: Translate Compound Statements Let p, q, and r represent the following: p: You get a promotion. q: You complete the training. r: You will receive a bonus. a. Write (p  𝒒) → r as an English sentence. b. Write “ If you do not complete the training, then you will not get a promotion and you will not receive a bonus” in symbolic form. Solution: a. Because the p and the q statements both appear in parentheses in the symbolic form, they are placed to the left of the comma in the English sentence. Thus, the translation is: If you get a promotion and complete the training, then you will receive a bonus. b. Because the not p and the not r statements are both to the right of the comma in the English sentence, they are grouped together in parentheses in the symbolic form. Thus the translation is: ~𝒒→(~p  ~𝒓) Check Your Progress 5 Let p, q, and r represent the following: p: Kesha’s singing style is similar to Uffie’s. q: Kesha has messy hair. r: Kesha is a rapper. a. Write (p  𝒒) → r as an English sentence. 6  b. Write “If Kesha is not a rapper, then Kesha does not have messy hair and Kesha’s singing style is not similar to Uffies’s.” in symbolic form. The use of parentheses in a symbolic statement may affect the meaning of the statement. For instance, ~(𝒑 𝒒) indicates the negation of the compound statement 𝒑 𝒒. However, ~𝒑 𝒒 indicates that only the 𝒑 statement is negated. The statement ~(𝒑 𝒒)is read as, “It is not true that, 𝒑 or 𝒒.”. The statement ~𝒑 𝒒 is read as, “Not 𝒑 or 𝒒. " If you order cake and ice cream in a restaurant, the waiter will bring both cake and ice cream. In general, the conjunction p  𝒒 is true if both p and 𝒒 are true, and the conjunction is false if either p or 𝒒 is false. Truth Value of a Conjunction The conjuction p  𝒒 is true if and only if both p and 𝑞 are true. The truth table below shows the four possible cases that arise when we form a conjunction of two statements. Truth Table for p  𝒒 p q p𝒒 T T T T F F F T F F F F Sometimes the word but is used in place of the connective and. For instance, “I ride my bike to school, but I ride the bus to work,” is equivalent to the conjunction. “I ride my bike to school and I ride the bus to work.” Any disjunction 𝒑 𝒒 is true if 𝒑 is true or 𝒒 is true or both 𝒑 and 𝒒 are true. Truth Value of a Disjunction The disjunction 𝒑 𝒒 is true if and only if 𝒑 is true , 𝒒 is true , or both 𝒑 and 𝒒 are true. The truth table below shows that the disjunction 𝒑 or 𝒒 is false if both 𝒑 and 𝒒 are false; however, it is true in all other cases. Truth Table for 𝒑 𝒒 p q 𝒑 𝒒 T T T T F T F T T F F F 7  Example 6: Determine the Truth Value of a Statement Determine whether each statement is true or false. a. 𝟕 ≥ 𝟓. b. 5 is a whole number and 5 is an even number. c. 2 is a prime number and 2 is an even number. Solution: a. 𝟕 ≥ 𝟓 means 7>5 or 7=5. Because 7>5is true, the statement 𝟕 ≥ 𝟓 is a true statement. b. This is a false statement because 5 is not an even number. c. This is a true statement because each simple statement is true. Check Your Progress 6 Determine whether each statement is true or false. a. 21 is a rational number and 21 is a natural number. b. 𝟒 ≤ 𝟗. c. −𝟕 ≥ −𝟑. 8  Activity 1 Answer the following. In numbers 1-5, determine whether each sentence is a statement. 1. Harvey Mudd college is in Oregon. 2. Have a fun trip. 3. Do you like to read? 4. 𝒙𝟐 = 𝟐𝟓 5. 𝒙=𝒙+𝟏 In numbers 6-10, write the negation of each statement. 6. The Giants lost game. 7. The lunch was served at noon. 8. The game did not go into overtime. 9. The game was not shown on ABC. 10. The tour goes to Singapore. In numbers 11-13, write each sentence in symbolic form. Represent each simple statement in the sentence with the letter indicated in the parentheses. Also state whether the sentence is a conjunction, a disjunction, a negation, a conditional, or a biconditional. 11. If today is Wednesday (w), then tomorrow is Thursday (t). 12. A triangle is an equilateral triangle (l) if and only if it is an equiangular triangle (a). 13. I will major in mathematics (m) or computer science (c). In numbers 14-16, write each symbolic statement in words. Use p, q, r, s, t, and u as defined below. p: The tour goes to Italy. q: The tour goes to Spain. r: We go to Venice. s: We to Florence. t: The hotel fees are included. u: The meals are not included. Activity 14. p1 ~𝒒 15. s →~𝒓 16. ~𝒕  𝒖 In numbers 17-20, determine whether each statement is true or false. 17. 7 < 5 or 3 > 1. 18. 𝟑 ≤ 𝟗. 19. −𝟓 ≥ −𝟏𝟏. 20. 𝟒. 𝟓 ≤ 𝟓. 𝟒. 9  Lesson 2: Truth Tables, Equivalent Statements, and Tautologies We consider methods of constructing truth tables for statement that involves a combination of conjunctions, disjunctions, and/or negations. If the given statement involves only two simple statements, then start with a table with four rows called the standard truth table form. Example 1: Truth Tables a. Construct a table for ~(~𝒑 𝒒) 𝒒. b. Use the truth table from part a to determine the truth value of ~(~𝒑 𝒒) 𝒒, given that p is true and q is false. Solution: a. Start with the standard truth table form and then include a ~𝒑 column. p q ~p T T F T F F F T T F F T Now use the truth values from the ~p and q columns to produce the truth values for ~p 𝒒, as shown in the rightmost column of the following table. p q ~p ~p 𝒒 T T F T T F F F F T T T F F T T Negate the truth values in the ~p 𝒒 column to produce the following. p q ~p ~p 𝒒 ~(~𝒑 𝒒) T T F T F T F F F T F T T T F F F T T F As our last step, we form the disjunction ~(~𝒑 𝒒) with q and place the results in the rightmost column of the table. See the following table. The shaded column is the truth table for ~(~𝒑 𝒒) 𝒒. 10  p q ~p ~p 𝒒 ~(~𝒑 𝒒) ~(~𝒑 𝒒) 𝒒 T T F T F T row 1 T F F F T T F T T row 2 T T F F F T T F F row 3 b. In row 2 of the above truth table, we see that when p is true, and q is false, therow 4 statement ~(~𝒑 𝒒) 𝒒 in the rightmost column is true. Check Your Progress 1 a. Construct a truth table for (𝒑 ~𝒒) (~𝒑𝒒). b. Use the truth table that you constructed in part a to determine the truth value (𝒑 ~𝒒) (~𝒑𝒒), given that p is true and q is false. Compound statements that involve exactly three simple statements require a standard truth table form with 𝟐𝒏 = 𝟖 rows, as shown below. Standard truth table form for a statement that involves the three simple statements p, q, r. p q r Given statement T T T T T F T F T T F F F T T F T F F F T F F F Example 2: Truth Tables a. Construct a truth table for (𝒑 𝒒) (~𝒓 𝒒). b. Use the truth table from part a to determine the truth value (𝒑 𝒒) (~𝒓 𝒒), given that p is true, q is true, and r is false. Solution: a. Using the procedures developed in Example 1, we can produce the following table. The shaded column is the truth table for (𝒑 𝒒) (~𝒓 𝒒). The numbers in the squares below the columns denote the order in which the columns were constructed. Each truth value in the column numbered 4 is the conjunction of the truth values to its left in the columns numbered 1 and 3. 11  p q r (𝒑 𝒒) ~𝒓 (~𝒓 𝒒) (𝒑 𝒒) (~𝒓 𝒒) T T T T F T T row 1 T T F T T T T row 2 T F T F F F F row 3 T F F F T T F row 4 F T T F F T F row 5 F T F F T T F row 6 F F T F F F F row 7 F F F F T T F row 8 1 2 3 4 b. In row 2 of the above truth table, we see that (𝒑 𝒒) (~𝒓 𝒒) is true when p is true, q is true, and r is false. Check your progress 2 a. Construct a truth table for (~𝒑 𝒓) (𝒒 ~𝒓). b. Use the truth table that you constructed in part a to determine the truth value of (~𝒑 𝒓) (𝒒 ~𝒓), given that p is false, q is true and r is false. Alternative Method for the Construction of a Truth Table In example 3 we use an alternative procedure to construct a truth table. Alternative Procedure for Constructing a Truth Table 1. If the given statement has n simple statements, then start with a standard form that has 2𝑛 rows. Enter the truth values for each simple statement and their negations. 2. Use the truth values for each simple statement and their negations to enter the truth values under each connective within a pair of grouping symbols, including parentheses (), brackets [ ], and braces { }. If some grouping symbols are nested inside other grouping symbols, then work from the inside out. In any situation in which grouping symbols have not been used, then we use the following order of procedure agreement. First assign the truth values to negations from left to right, followed by conjunctions from left to right, followed by disjunction from left to right, followed by conditionals from left to right, and finally by biconditionals from left to right. 3. The truth values that are entered into the column under the connective for which truth values are assigned last, form the truth table for the given statement. 12  Example 3: Use the Alternative Procedure to Construct a Truth Table Construct a truth table 𝒑 [~(𝒑~𝒒)]. Solution: Step 1: The given statement 𝒑 [~(𝒑~𝒒)] has the two simple statements p and q. thus we start with a standard form that has 𝟐𝒏 = 𝒓𝒐𝒘𝒔. In each column, enter the truth values for the statements p and ~𝒒, as shown in the columns numbered 1, 2, and 3 of the following table. p q 𝒑  [~ (𝒑  ~𝒒)] T T T T F T F T T T F T F F F F F F F T 1 2 3 Step 2: Use the truth values in columns 2 and 3 to determine the truth values to enter under the “and” connective. See column 4 in the following truth table. Now negate the truth values in column 4 to produce the truth values in column 5. p q 𝒑  [~ (𝒑  ~𝒒)] T T T T T F F T F T F T T T F T F T F F F F F F T F F T 1 5 2 4 3 Step 3: Use the truth values in the columns 1 and 5 to determine the truth values to enter under the “or” connective. See column 6 in the following table. Shaded column 6 is the truth table for 𝒑 [~(𝒑~𝒒)]. p q 𝒑  [~ (𝒑  ~𝒒)] T T T T T T F F T F T T F T T T F T F T T F F F F F F T T F F T 1 6 5 2 4 3 Check Your Progress Construct a truth table for ~𝒑(𝒑 𝒒). 13  Equivalent statements Two statements are equivalent if they both have the same truth value for all possible truth values of their simple statements. Equivalent statements have identical truth values in the final columns of their truth tables. The notation 𝑝 ≡ 𝑞 is used to indicate that the statements p and q are equivalent. Example 4: Verify That Two Statements Are Equivalent Show that ~(𝒑 ~𝒒) and ~𝒑  𝒒 are equivalent statements. Solution: Construct two truth tables and compare the results. The truth tables below that ~(𝒑 ~𝒒) and ~𝒑  𝒒 have the same truth values for all possible truth values of their simple statements. Thus the statements are equivalent. p q ~ (𝒑  ~𝒒) p q ~𝒑  𝒒 T T F T T F T T F F T T F F T T T T F F F F F T T F F F F T T T T F F F F T T F F F F F 4 1 3 2 1 3 2 identical truth values Thus ~(𝒑 ~𝒒) ≡ ~𝒑  𝒒. Check Your Progress 4 Show that 𝒑 (𝒑 ~𝒒) and 𝒑 are equivalent. Tautologies and Self-Contradictions A tautology is a statement that is always true. A self-contradiction is a statement that is always false. Example 6: Verify Tautologies and Self-Contradictions Show that 𝒑 (~𝒑 𝒒) is a tautology. 14 Solution: Enter the truth values for each simple statement and its negation as shown in the columns numbered 1, 2 and 3. Use the truth values in columns 2 and 3 to determine the truth values to enter in column 4, under the “or” connective. Use the truth values in columns 1 and 4 to determine the truth values to enter in column 5, under the “or” connective. p q 𝒑  (~𝒑  𝒒) T T T T F T T T F T T F F F F T F T T T T F F F T T T F 1 5 2 4 3 Column 5 of the table shows that 𝒑 (~𝒑 𝒒) is always true. Thus 𝒑 (~𝒑 𝒒) is tautology. Check Your Progress 6 Show that 𝒑 (~𝒑  𝒒) is a self-contradiction. Activity 2 Answer the following. In numbers 1-3, determine the truth value of the compound statement given that p is a false statement, q is a true statement, and r is a true statement. 1. 𝒑 (~𝒒 𝒓) 2. 𝒓 ~(𝒑 𝒓) 3. (𝒑 𝒒) (~𝒑 ~𝐪) In numbers 4-6, construct a truth table for each compound statement. 4. ~𝒑 𝒒 5. (𝒒 ~𝒑) ~𝒒 6. 𝒑[~(𝒑~𝒒)] In numbers 7-9, use two truth tables to show that each of the statements are equivalent. 7. 𝒑 (𝒑 𝒓), 𝒑 8. 𝒒 (𝒒 𝒓), 𝒒 9. 𝒑 (𝒒 𝒓), (𝒑𝒒)(𝒑𝒓) In numbers 10-12, use a truth table to determine whether the given statement is tautology. 10. 𝒑 ~𝒑 11. 𝒒[~(𝒒𝐫)~𝒒] 12. (𝒑 𝒒) (~𝒑 𝒒) In numbers 13-15, use a truth table to determine whether the given statement is a self-contradiction. 13. ~𝒓  𝒓 14. ~(𝒑 ~𝒑) 15. 𝒑 (~𝒑  𝒒) 15  Lesson 3: Conditional, Biconditional, and Related Statements Conditional Statements Conditional Statements can be written in if p, then q form or in if p, q form. For instance, all of the following are conditional statements. If we order pizza, then we can have it delivered. If you go to the movie, you will not be able to meet us for dinner. If n is a prime number greater than 2, then n is an odd number. In any conditional statement represented by “If p, then q”, the p statement is called the antecedent and the q statement is called the consequent. Example 1: Identify the Antecedent and Consequent of a Conditional Identify the antecedent and consequent in the following statements. a. If our school was this nice, I would go there more than once a week. -The Basketball Diaries b. If you don’t get in the plane, you’ll regret it. -Casablanca c. If you strike me down, I shall become more powerful than you can possibly imagine. -Obi-Wan Kenobi, Star Wars, Episode IV, A New Hope Solution: a. Antecedent: our school was this nice Consequent: I would go there more than once a week b. Antecedent: you don’t get in the plane Consequent: you’ll regret it c. Antecedent: you strike me down Consequent: I shall become more powerful than you can possibly imagine Check Your Progress 1 Identify the antecedent and consequent in each of the following conditional statements. a. If I study for at least 6 hours, then I will get an A on the test. b. If I get the job, I will buy a new car. c. If you can dream it, you can do it. Arrow Notation The conditional statement, “If p, then q”, can be written using the arrow notation p→ q. The arrow notation p→ q is read as “if p, then q” or as “p implies q”. 16 The Truth Table for the Conditional p→ q Truth Value of the Conditional p→ q The conditional p→ q is false if p is true and q is false. It is true in all other cases. Truth Table for p→ q p q p→𝒒 T T T T F F F T T F F T Example 2: Find the Truth Value of a Conditional Determine the truth value of each of the following conditional statements. a. If 2 is an integer, then 2 is a rational number. b. If 3 is a negative number, then 5>7. c. If 5>3, then 2+7=4. Solution: a. Because the consequent is true, this is a true statement. b. Because the antecedent is false, this is a true statement. c. Because the antecedent is true and the consequent is false, this is a false statement. Check Your Progress 2 Determine the truth value of each of the following. a. If 𝟒 ≥ 𝟑, the 2 + 5 = 6. b. If 5>9, then 4>9. c. If Tuesday follows Monday, then April follows March. Example 3: Construct a Truth Table for a Statement Involving a Conditional Construct a truth table for [𝒑(𝒒~𝒑)]→~𝒑. Enter the truth values for each simple statement and its negation as shown in column 1, 2, 3, and 4. Use the truth values in columns 2 and 3 to determine the truth values to enter in column 5, under the “or” connective. Use the truth values in columns 1 and 5 to determine the truth values to enter in column 6 under the “and” connective. Use the truth values in columns 6 and 4 to determine the truth values to enter in column 7 under the “If… then” connective. 17  p q [𝒑  (𝒒  ~𝒑)] → ~𝒑 T T T T T T F F F T F T F F F F T F F T F F T T T T T F F F F F T T T T 1 6 2 5 3 7 4 Check Your Progress 3 Construct a truth table for [𝒑(𝒑→𝒒)]→𝒒. An Equivalent Form of the Conditional An equivalent Form of the Conditional 𝒑→𝒒 𝒑→𝒒 ≡ ~𝒑  𝒒 Example 4: Write a Conditional in its Equivalent Disjunctive Form Write each of the following in its equivalent disjunctive form. a. If I could play the guitar, I would join the band. b. If Cam Newton cannot play, then his team will lose. Solution: In each case we write the disjunction of the negation of the antecedent and the consequent. a. I cannot play the guitar or I would join the band. b. Cam Newton can play or his team will lose. Check Your Progress 4. Write each of the following in its equivalent disjunctive form. a. If I don’t move to Georgia, I will live in Houston. b. If the number is divisible by 2, then the number is even. The Negation of the Conditional Because 𝒑→𝒒 ≡ ~𝒑  𝒒, an equivalent form of ~(𝒑→𝒒) is given by ~(~𝒑  𝒒), which, by one of the De Morgan’s laws, can be expressed as the conjunction 𝒑 ~ 𝒒. The Negation of 𝒑→𝒒 ~ (𝒑→𝒒) ≡ 𝒑 ~ 𝒒 18  Example 5: Write the Negation of a Conditional Statement Write the negation of each conditional statement. a. If they pay me the money, I will sign the contract. b. If the lines are parallel, then they do not intersect. Solution: In each case, we write the conjunction of the antecedent and the negation of the consequent. a. They paid me the money and I did not sign the contract. b. The lines are parallel and they intersect. Check Your Progress 5 Write the negation of each conditional statement. a. If I finish the report, I will go to the concert. b. If the square of n is 25, then n is 5 or -5. The Biconditional The statement (𝒑→𝒒) (𝒒→𝒑) is called a biconditional and is denoted by 𝒑 ↔q, which is read as “p if and only if q”. The Biconditional 𝒑 ↔ 𝒒 𝑝 ↔ 𝑞 ≡ [(𝑝→𝑞) (𝑞→𝑝) ] Example 6: Write Symbolic Biconditional Statements in Words Let p, q, and r represents the following: p: She will go on vacation. q: She cannot take the train. r: She cannot get a loan. Write the following symbolic statements in words. a. 𝒑 ↔ ~𝒒 b. ~𝐫 ↔ ~𝒑 Solution: a. She will go on vacation if and only if she can take the train. b. She can get a loan if and only if she does not go on vacation. Check Your Progress 6 Use p, q, and r as defined in Example 6. Write the following symbolic statements in words. a. 𝒑 ↔ ~𝒓 b. ~𝒒 ↔ ~𝒓 19 Table below shows that 𝒑 ↔ 𝒒 is true only when p and q have the same truth value. Truth Table for 𝒑 ↔ 𝒒 p q p↔ 𝒒 T T T T F F F T F F F T Example 7: Determine the Truth Value of a Biconditional State whether each biconditional is true or false. a. x+4=7 if and only if x=3. b. 𝒙𝟐 =36 if and only if x=6. Solution: a. Both equations are true when x=3, and both are false when x ≠3. Both equations have the same truth value for any given value of x, so this is a true statement. b. If x=-6, the first equation is true and the second equation is false. Thus this is a false statement. Check Your Progress 7 State whether each biconditional is true or false. a. x > 7 if and only if x > 6. b. x + 5 > 7 if and only if x > 2. The Conditional and Related Statements The Converse, the Inverse, and the Contrapositive Every conditional statement has three related statements. They are called the converse, the inverse, and the contrapositive. Statements Related to the Conditional Statement The converse of 𝑝→𝑞 is 𝑞→𝑝. The inverse of 𝑝→𝑞 is ~𝑝→~𝑞. The contrapositive of 𝑝→𝑞 is ~𝑞→~𝑝. The above definitions show the following: ▪ The converse of 𝒑→𝒒 is formed by interchanging the antecedent p with the consequent q. ▪ The inverse of 𝒑→𝒒 is formed by negating the antecedent p and negating the consequent q. ▪ The contrapositive of 𝒑→𝒒 is formed by negating both the antecedent p and the consequent q and interchanging these negated statements. 20  Example 1: Write the Converse, Inverse, and Contrapositive of a Conditional Write the converse, inverse, and contrapositive of If I get the job, then I will rent the apartment. Solution: Converse: If I rent the apartment, then I get the job. Inverse: If I do not get the job, then I will not rent the apartment. Contrapositive: If I do not rent the apartment, then I did not get the job. Check Your Progress 1 Write the converse, inverse, and contrapositive of If we have a quiz today, then we will not have a quiz tomorrow. Truth Tables for Conditional and Related Statements Example 2: Determine Whether Related Statements Are Equivalent Determine whether the given statements are equivalent. a. If a number ends with a 5, then the number is divisible by 5. If a number is divisible by 5, then the number ends with a 5. b. If two lines in a plane do not intersect, then the lines are parallel. If two lines in a plane are not parallel, then the lines intersect. Solution: a. The second statement is the converse of the first. The statements are not equivalent. b. The second statement is the contrapositive of the first. The statements are equivalent. 21  Check Your Progress 2 Determine whether the given statements are equivalent. a. If a = b, then a · c = b · c. If a ≠ b, then a · c ≠ b · c. b. If I live in Nashville, then I live in Tennessee. If I do not live in Tennessee, then I do not live in Nashville. In mathematics, it is often necessary to prove statements that are in “If p, then q” form. If a proof cannot be readily produced, mathematicians often try to prove the contrapositive “If ~𝒒 𝐭𝐡𝐞𝐧 ~𝒑”. Because conditional and its contrapositive are equivalent statements, a proof of either statement also establishes the proof of the other statement. Example 3: Use the Contrapositive to Determine a Truth Value Write the contrapositive of each statement and use the contrapositive to determine whether the original statement is true or false. a. If a + b is not divisible by 5, then a and b are not both divisible by 5. b. If 𝒙𝟑 is an odd integer, then x is an odd integer. (Assume x is an integer.) c. If a geometric figure is not a rectangle, then it is not a square. Solution: a. If a and b are both divisible by 5, then a+b is divisible by 5. This is a true statement, so the original statement is also true. b. If x is an even integer, then 𝒙𝟑 is an even integer. This is a true statement, so the original statement is also true. c. If a geometric figure is a square, then it is a rectangle. This is a true statement, so the original statement is also true. Check Your Progress 3 Write the contrapositive of each statement and use the contrapositive to determine whether the original statement is true or false. a. If 3 + x is an odd integer, then x is an even integer. (Assume x is an integer.) b. If two triangles are not similar triangles, then they are not congruent triangles. Note: Similar triangles have the same shape. Congruent triangles have the same size and shape. c. If today is not Wednesday, then tomorrow is not Thursday. 22 Activity 3 Answer the following. In numbers 1-5, identify the antecedent and the consequent. 1. If I had the money, I would buy the painting. 2. If Shelly goes on the trip, she will not be able to take part in the graduation ceremony. 3. If they had a guard log, then no one would trespass on their property. 4. If I don’t get to school before 7:30, I won’t be able to find a parking place. 5. If I change my major, I must reapply for admission. In numbers 6-8, determine the truth value of the given statement. 6. If x is an even integer, then 𝒙𝟐 is an even integer. 7. If x is a prime number, then x+2 is a prime number. 8. All frogs can dance, then today is Monday. In numbers 9-12, write each sentence in symbolic form, Use v, p, and t as defined below. 9. I will take a vacation if and only if I get the promotion. 10. If I get the promotion, I will take a vacation. 11. If I am transferred, then I will not take a vacation. 12. If I will not take a vacation, then I will not be transferred and I get the promotion. In numbers 13-15, construct a truth table for each statement to determine if the statements are equivalent. 13. 𝒑→~𝒓, 𝒓~𝒑 14. 𝒑→𝒒, 𝒒→𝒑 15. ~𝒑→(𝒑𝒓), 𝐫 23  POST TEST A. State whether each of the following sentences is a statement or not. If it is a statement, determine its truth value. If it is not a statement, give your reason. 1. Is it raining? 2. 2016 is a leap year. 3. The sum of two prime numbers is even. 4. Cagayan de Oro City is located in Mindanao, Philippines 5. Shut the door! B. Find the solution set for each of the following open sentences. 1. 5x+4 = 19 2. 4 –y < 5 3. 4x > x + 9 4. n2 – n = 6 5. x + y = 5 C. Determine the truth value of each of the following statements. 1. If March is the third calendar month, then one foot has 12 inches 2. March is the third calendar month implies one foot has more than 12 inches. 3. Two is a prime number if one is an integer 4. If 2018 is a leap year, then February will have 29 days. 5. 2018 is a leap year if and only if February will have 20 days. D. Create your own tessellation. You can choose any images you want to use in your tessellation. Use a bond paper. REFERENCES Books 1. Hengania, C., et. al. (2018). Mathematics in the Modern World. Malabon City: Mutya Publishing House Inc. 2. Cenage. (2018). Mathematics in the Modern World. Manila: Rex Book Store, Inc 3. Balatazar, E. (2018). Mathematics in the Modern World. Quezon City: C & E Publishing, Inc. 4. Alejan, R. (2018). Mathematics in the Modern World. Malabon City: Mutya Publishing House, Inc. 24