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![Truth Tables (via Order of Precedence)
Construct a truth table for 𝑝 ∨ [~ 𝑝 ∧ ~𝑞 ].](https://image.slidesharecdn.com/1-220917072338-0c496726/85/1-Logic-pptx-18-320.jpg)
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1. Logic.pptx
- 1. ELEMENTARY LOGIC Davao Oriental StateUniversity Graduate School Department
- 2. Logic Statements Every languagecontains different types of sentences, such as statements, questions, and commands. For instance, • “Is the test today?” is a question. • “Go get the newspaper” is a command. • “This is a nice car” is an opinion. • “Manila is the capital of Philippines” is a statement of fact.
- 3. Statement A statement isa declarative sentence that is either true or false, but not both true and false. • It may not be necessary to determine whether a sentence is true to determine whether it is a statement. For instance, consider the following sentence. American Shaun White won an Olympic gold medal in speed skating. • You may not know if the sentence is true, but you do know that the sentence is either true or it is false, and that it is not both true and false. Thus, you know that the sentence is a statement.
- 4. Let’s try! Determine whethereach sentence is a statement. a. Lupon is a municipality in Davao Oriental b. How are you? c. 99 + 2 is a prime number. d. 𝑥 + 1 = 5. a. Lupon is a municipality in Davao Oriental, so the sentence is true and it is a statement. b. The sentence is a question; it is not a declarative sentence. Thus it is not a statement. c. You may not know whether 99 + 2 is a prime number; however, you do know that it is a whole number larger than 1, so it is either a prime number or it is not a prime number. The sentence is either true or it is false, and it is not both true and false, so it is a statement. d. 𝑥 + 1 = 5 is a statement. It is known as an open statement. It is true for 𝑥 = 4, and it is false for any other values of 𝑥. For any given value of 𝑥, it is true or false but not both.
- 5. Simple Statements andCompound Statements A simple statement is a statement that conveys a single idea. A compound statement is a statement that conveys two or more ideas. Connecting simple statements with words and phrases such as and, or, if . . . then, and if and only if creates a compound statement. | Example: “I will attend the meeting or I will go to school.” is a compound statement.
- 6. Logic Connectives andSymbols
- 7. Let’s try! Consider thefollowing 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. 𝒑 ∧ 𝒒 ~𝒒 ∧ 𝒓 ~𝒔 ∨ 𝒓 𝒒 → 𝒔
- 8. Let’s try! Consider thefollowing simple statements. p: The game will be played in Araneta. q: The game will be shown on ABS. r: The game will not be shown in GMA. s: The Raptors are favored to win. Write the following symbolic statements in words. a. 𝒒 ∧ 𝒑 b. ~𝒓 ∧ 𝒔 c. 𝒔 ↔ ~𝒑 The game will be shown on ABS and the game will be played in Araneta. The game will be shown on GMA and the Raptors are favored to win. The Raptors are favored to win if and only if the game will not be played in Araneta.
- 9. Compound Statements andGrouping Symbols If a compound statement is written in symbolic form, then parentheses are used to indicate which simple statements are grouped together.
- 10. Compound Statements andGrouping Symbols 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
- 11. Let’s try! 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. If you get a promotion and you complete the training, then you will receive a bonus. 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. ~𝒒 → (~𝒑 ∧ ~𝒓)
- 12. Truth Value andTruth 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.
- 13. Truth Value andTruth Tables 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.
- 14. Truth Value ofa Conjunction The conjunction 𝑝 ∧ 𝑞 is true if and only if both 𝑝 and 𝑞 are true.
- 15. Truth Value ofa Disjunction The disjunction 𝑝 ∨ 𝑞 is true if and only if 𝑝 is true, 𝑞 is true, or both 𝑝 and 𝑞 are true.
- 16. Truth Value andTruth Tables Construct a truth table for ~ ~𝑝 ∨ 𝑞 ∨ 𝑞. 𝑝 𝑞 T T T F F T F F ~𝑝 F F T T ~𝑝 ∨ 𝑞 T F T T ~(~𝑝 ∨ 𝑞) F T F F ~(~𝑝 ∨ 𝑞) ∨ 𝑞 T T T F
- 17. Truth Value andTruth Tables Construct a truth table for 𝑝 ∧ 𝑞 ∧ (~𝑟 ∨ 𝑞). 𝑝 𝑞 𝑟 T T T T T F T F T T F F F T T F T F F F T F F F 𝑝 ∧ 𝑞 T T F F F F F F ~𝑟 F T F T F T F T ~𝑟 ∨ 𝑞 T T F T T T F T 𝑝 ∧ 𝑞 ∧ (~𝑟 ∨ 𝑞) T T F F F F F F
- 18. Truth Tables (viaOrder of Precedence) Construct a truth table for 𝑝 ∨ [~ 𝑝 ∧ ~𝑞 ].
- 19. Equivalent Statements Two statementsare 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.
- 20. Equivalent Statements Show that~(𝑝 ∨ ~𝑞) and ~𝑝 ∧ 𝑞 are equivalent statements.
- 21. De Morgan’s Law Forany statements p and q De Morgan’s law can be used to restate certain English sentence in its equivalent form
- 22. De Morgan’s Law Useone of De Morgan’s laws to restate the following sentence in an equivalent form. It is not true that, I graduated or I got a job. Let p represent the statement “I graduated” and let q represent the statement “I got a job”. By ~ 𝑝 ∨ 𝑞 ≡ ~𝑝 ∧ ~𝑞, we have, I did not graduate and I did not get a job
- 23. Tautologies and Self-Contradictions Atautology is a statement that is always true. A self-contradiction is a statement that is always false.
- 24. Tautologies and Self-Contradictions Showthat 𝑝 ∨ (~𝑝 ∨ 𝑞) is a tautology.
- 25. Tautologies and Self-Contradictions Showthat 𝑝 ∧ (~𝑝 ∧ 𝑞) is a self-contradiction.
- 26. Conditional Statements Conditional statementscan 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” or by “If p, q,” the p statement is called the antecedent and the q statement is called the consequent. The conditional statement, “If p, then q,” can be written using the arrow notation 𝑝 → 𝑞. The arrow notation is read as “if p, then q” or as “p implies q.” Institute of Computng and Engineering Mathematics Department | Jerd M. Dela Gente, MSc.
- 27. Conditional Statements Identify theantecedent and consequent in the following statements. a. If our school was this nice, I would go there more than once a week. b. If you don’t stop and look around once in a while, you could miss it. c. If you strike me down, I shall become more powerful than you can possibly imagine. a. Antecedent: our school was this nice Consequent: I would go there more than once a week b. Antecedent: you don’t stop and look around once in a while Consequent: you could miss it c. Antecedent: you strike me down Consequent: I shall become more powerful than you can possibly imagine Institute of Computing and Engineering Mathematics Department | Jerd M. Dela Gente, MSc.
- 28. Truth Value andTruth Table for 𝒑 → 𝒒 The conditional 𝑝 → 𝑞 is false if p is true and q is false. It is true in all other cases.
- 29. Truth Value andTruth Table for 𝒑 → 𝒒 Determine the truth value of each of the following. 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. 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.
- 30. Truth Value andTruth Table for 𝒑 → 𝒒 Construct a truth table for 𝑝 ∧ 𝑞 ∨ ~𝑝 → ~𝑝.
- 31. An Equivalent Formof the Conditional The conditional 𝑝 → 𝑞 is equivalent to the disjunction ~𝑝 ∨ 𝑞 𝑝 → 𝑞 ≡ ~𝑝 ∨ 𝑞 Example: If I could play the guitar, I would join the band. I cannot play the guitar or I would join the band.
- 32. Negation of theConditional Because 𝑝 → 𝑞 ≡ ~𝑝 ∨ 𝑞, an equivalent form of ~(𝑝 → 𝑞) is given by ~(~𝑝 ∨ 𝑞), which, by one of De Morgan’s laws, can be expressed as the conjunction 𝑝 ∧ ~𝑞. Example: 𝑝 → 𝑞 :If they pay me the money, I will sign the contract. ~ 𝑝 → 𝑞 : They paid me the money and I did not sign the contract. ~ 𝑝 → 𝑞 ≡ 𝑝 ∧ ~𝑞
- 33. 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.” 𝑝 ↔ 𝑞 ≡ (𝑝 → 𝑞) ∧ (𝑞 → 𝑝)
- 34. Truth Value andTruth Table for 𝒑 ↔ 𝒒 The biconditional 𝑝 ↔ 𝑞 is true only when p and q have the same truth value.
- 35. Truth Value andTruth Table for 𝒑 ↔ 𝒒 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. a. Both equations are true when 𝑥 = 3, and both are false when 𝑥 ≠ 3. Both equations have the same truth value for any 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.
- 36. Symbolic Arguments An argumentconsists of a set of statements called premises and another statement called the conclusion. An argument is valid if the conclusion is true whenever all the premises are assumed to be true. An argument is invalid if it is not a valid argument.
- 37. Example: If Aristotle washuman, then Aristotle was mortal. Aristotle was human. Therefore, Aristotle was mortal. The argument has two premises and 1 conclusion: First premise: If Aristotle was human, then Aristotle was mortal. Second premise: Aristotle was human. . Conclusion: Therefore, Aristotle was mortal. Let ℎ represent the statement “Aristotle was human” and let 𝑚 represent the statement “Aristotle was mortal” The argument in symbolic form is, ℎ → 𝑚 ℎ ∴ 𝑚 Symbolic Arguments The three dots are a symbol for “therefore.”
- 38. Example: The fish isfresh or I will not order it. The fish is fresh. Therefore I will order it. Let 𝑓 represent the statement “The fish is fresh” and let 𝑜 represent the statement “I will order it” The argument in symbolic form is, 𝑓 ∨ ~𝑜 𝑓 ∴ 𝑜 Symbolic Arguments
- 39. Arguments and TruthTable Validity of the Aristotle example If the conclusion is true in every row of the truth table in which all the premises are true, the argument is valid. If the conclusion is false in any row in which all of the premises are true, the argument is invalid. Row 1 is the only row in which all the premises are true, so it is the only row that we examine. Because the conclusion is true in row 1, the argument is valid.
- 40. Arguments and TruthTable Determine whether the following argument is valid or invalid. If it rains, then the game will not be played. It is not raining. Therefore, the game will be played. Because the conclusion in row 4 is false and the premises are both true, the argument is invalid.
- 41. Arguments and TruthTable Determine whether the following argument is valid or invalid. Because the conclusion is true in the rows 2, 6, 7 and 8, the argument is valid.
- 42. Standard Forms ofValid Arguments
- 43. Standard Forms ofInvalid Arguments
- 44. Validity of argumentexamples Use a standard form to determine whether the following argument is valid or invalid. This symbolic form matches the standard form known as disjunctive syllogism. Thus the argument is valid.
- 45. Validity of argumentexamples Consider an argument with the following symbolic form. This sequence of valid arguments shows that t is a valid conclusion for the original argument.
- 46. Validity of argumentexamples Determine whether the following argument is valid. If the movie was directed by Steven Spielberg (𝑠), then I want to see it (𝑤). The movie’s production costs must exceed $50 million (𝑐) or I do not want to see it. The movie’s production costs were less than $50 million. Therefore, the movie was not directed by Steven Spielberg. This sequence of valid arguments shows that t is a valid conclusion for the original argument.