2/13/2012. Logic: Truth Tables. CS160 Rosen Chapter 1. Logic?
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1 Logic: Truth Tables CS160 Rosen Chapter 1 Logic? 1
2 What is logic? Logic is a truth-preserving system of inference Truth-preserving: If the initial statements are true, the inferred statements will be true System: a set of mechanistic transformations, based on syntax alone Inference: the process of deriving (inferring) new statements from old statements Propositional Logic A proposition is a statement that is either true or false Examples: This class is CS160 (true) Today is Sunday (false) It is currently raining in Singapore (???) Every proposition is true or false, but its truth value (true or false) may be unknown 2
3 Propositional Logic (II) A propositional statement is one of: A simple proposition denoted by a capital letter, e.g. A. A negation of a propositional statement e.g. A : not A Two propositional statements joined by a connective e.g. A B : A and B e.g. A B : A or B If a connective joins complex statements, parenthesis are added e.g. A (B C) Truth Tables The truth value of a compound propositional statement is determined by its truth table Truth tables define the truth value of a connective for every possible truth value of its terms 3
4 Logical negation Negation of proposition A is A A: It is snowing. A: It is not snowing A: Newton knew Einstein. A: Newton did not know Einstein. A: I am not registered for CS195. A: I am registered for CS195. Negation Truth Table A T F A F T 4
5 Logical and (conjunction) Conjunction of A and B is A B A: CS160 teaches logic. B: CS160 teaches Java. A B: CS160 teaches logic and Java. Combining conjunction and negation A: I like fish. B: I like sushi. I like fish but not sushi: A B Truth Table for Conjunction A B A B T T T T F F F T F F F F 5
6 Logical or (disjunction) Disjunction of A and B is A B A: Today is Friday. B: It is snowing. A B: Today is Friday or it is snowing. This statement is true if any of the following hold: Today is Friday It is snowing Both Otherwise it is false Truth Table for Disjunction A B A B T T T T F T F T T F F F 6
7 Exclusive Or The or connective is inclusive: it is true if either or both arguments are true There is also an exclusive or A B A B T T F T F T F T T F F F Confusion over Inclusive OR and Exclusive OR Restaurants typically let you pick one (either soup or salad, not both) when they say The entrée comes with a soup or salad. Use exclusive OR to write as a logic proposition Give two interpretations of the sentence using inclusive OR and exclusive OR: Students who have taken calculus or intro to programming can take this class 7
8 Unidirectional implication Shown by A B A: A programming homework is due. B: It is Tuesday. A B: If a programming homework is due, then it must be Tuesday. A programming homework is due only if it is Tuesday. Is this the same as: If it is Tuesday, then a programming homework is due. Unidirectional implication and other connectives A: You can access free Internet at this hotel. B: You are attending a conference. C: You are staying at this hotel. You can access free internet at this hotel only if you are attending a conference or you are staying at this hotel. A ( B C ) What if we said if instead of only if? 8
9 Unidirectional implication and other connectives (contd) A: You can register for CS517. B: You have taken CS414. C: Your instructor gave an override. You cannot register for CS517 if you haven t taken CS414 unless your instructor gave an override. ( B C ) A Equivalent: A B C) Unidirectional & Bidirectional Implication The unidirectional implication connective is The bidirectional implication connective is These, too, are defined by truth tables A B A B T T T T F F F T T F F T A B A B T T T T F F F T F F F T 9
10 Bi conditional A: You can drive a car. B: You have a driver s license. You can drive a car if and only if you have a driver s license. A B What if we said if? What if we said only if? Compound Truth Tables Truth tables can also be used to determine the truth values of compound statements, such as (A B) ( A) (fill this as an exercise) A B A A B (A B) ( A) T T F T F T F F T F F T T T T F F T F F 10
11 Proofs of Equivalence Truth tables can be used to show the equivalence of two logical expressions The last two columns have the same values -- so the expressions are equivalent A B A B (A B) A B T T F T T T F T F F F T T F F F F F T T = Tautology and Contradiction A tautology is a compound proposition that is always true. A contradiction is a compound proposition that is always false. A contingency is neither a tautology nor a contradiction. A compound proposition is satisfiable if there is at least one assignment of truth values to the variables that makes the statement true. 11
12 Examples A A A A A A T F T F F T T F Result is always true, no matter what A is Therefore, it is a tautology Result is always false, no matter what A is Therefore, it is a contradiction Logical Proof Given a set of axioms Statements asserted to be true Prove a conclusion Another propositional statement In other words: Show that the conclusion is true whenever the axioms are true 12
13 Proof Method #1: Truth Tables Write out the truth tables for the axioms and conclusion The conclusion follows from the axioms if (and only if): The consequent is true for every row in which all the axioms are true Proof by Truth Table: Example Prove (A B), given (A B) A B A B A B T T T T T F F T F T F T F F F F Only row for which axiom is true Conclusion is true, so it follows from the axioms 13
14 Logical Equivalence Two compound propositions, p and q, are logically equivalent if p q is a tautology. Notation: p q De Morgan s Laws: (p q) p q (p q) p q How so? Let s build a truth table! Prove (p q) p q p q p q (p q) (p q) p q T T F F T F F T F F T F T T F T T F F T T F F T T F T T = 14
15 Show (p q) p q p q p q p q (p q) p q T T F F T F F T F F T T F F F T T F T F F F F T T F T T = Other Equivalences Show p q p q Show Distributive Law: p (q r) (p q) (p r) 15
16 Show p q p q p q p p q p q T T F T T T F F F F F T T T T F F T T T = Show p (q r) (p q) (p r) p q r q r p q p r p (q r) (p q) (p r) F F F F F F F F F F T F F T F F F T F F T F F F F T T T T T T T T F F F T T T T T F T F T T T T T T F F T T T T T T T T T T T T = 16
17 More Equivalences Equivalence p T p p F p p q q p p q q p p (p q) p p (p q) p Name Identity Commutative Absorption See Rosen for more. Equivalences with Conditionals and Biconditionals Conditionals p q p q p q q p (p q) p q Biconditionals p q (p q) (q p) p q p q (p q) p q 17
18 Prove Biconditional Equivalence p q p q (p q) p q T T T F F T F F T T F T F T T F F T F F = Proof By Truth Table If the consequent is true in the truth table whenever the axioms are true, it is proved Warning: when the axioms are false, the consequent may be true or false Problem: given n propositions, the truth table has 2 n rows Proof by truth table quickly becomes infeasible 18
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