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Chapter 1: The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy 1 1.1: Propositional Logic Propositions: A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. 2 Example 1: All the following declarative sentences are propositions: 1. Washington D.C., is the capital of the USA. 2. Toronto is the capital of Canada 3. 1+1=2. 4. 2+2=3. 3 Example 2: Consider the following sentences. Are they propositions? 1. What time is it? 2. Read this carefully. 3. x+1=2. 4. x+y=z 4 • We use letters to denote propositional variables (or statement variables). • T: the value of a proposition is true. • F: the value of a proposition is false. • The area of logic that deals with propositions is called the propositional calculus or propositional logic. 5 Let p and q are propositions: Definition 1: Negation (Not) • Symbol: ¬ • Statement: “it is not the case that p”. • Example: P: I am going to town ¬P: It is not the case that I am going to town; I am not going to town; I ain’t goin’. 6 7 Definition 2: Conjunction (And) • Symbol: • The conjunction pq is true when both p and q are true and is false otherwise. • Example: P - ‘I am going to town’ Q - ‘It is going to rain’ PQ: ‘I am going to town and it is going to rain.’ 8 t01_1_002.jpg 9 Definition 3: Disjunction (Or) • Symbol: • The disjunction pq is false when both p and q are false and is true otherwise. • Example: P - ‘I am going to town’ Q - ‘It is going to rain’ P Q: ‘I am going to town or it is going to rain.’ 10 11 Definition 4: Exclusive OR • Symbol: • The exclusive or of p and q, denote pq, is true when exactly one of p and q is true and is false otherwise. • Example: P - ‘I am going to town’ Q - ‘It is going to rain’ P Q: ‘Either I am going to town or it is going to rain.’ 12 13 Definition 5: Implication • • • • • If…. Then…. Symbol: The conditional statement pq is false when p is true and q is false, and true P is called the hypothesis and q is called the conclusion. Example: P - ‘I am going to town’ Q - ‘It is going to rain’ P Q: ‘If I am going to town then it is going to rain.’ 14 15 Equivalent Forms If P, then Q P implies Q If P, Q P only if Q P is a sufficient condition for Q Q if P Q whenever P Q is a necessary condition for P 16 Note: The implication is false only when P is true and Q is false! • ‘If the moon is made of green cheese then I have more money than Bill Gates’ (?) • ‘If the moon is made of green cheese then I’m on welfare’ (?) • ‘If 1+1=3 then your grandma wears combat boots’ (?) • ‘If I’m wealthy then the moon is not made of green cheese.’ (?) • ‘If I’m not wealthy then the moon is not made of green cheese.’ (?) 17 More terminology •QP is the CONVERSE of P Q •¬ Q ¬ P is the CONTRAPOSITIVE of P Q •¬ P ¬ Q is the inverse of P Q •Example: Find the converse of the following statement: R: ‘Raining tomorrow is a sufficient condition for my not going to town.’ 18 Procedure Step 1: Assign propositional variables to component propositions P: It will rain tomorrow Q: I will not go to town Step 2: Symbolize the assertion R: P Q Step 3: Symbolize the converse QP Step 4: Convert the symbols back into words ‘If I don’t go to town then it will rain tomorrow’ • Homework: Find inverse and contrapositive of statements above. 19 Definition 6: Biconditional • • • • • ‘if and only if’, ‘iff’ Symbol: The biconditional statement pq is true when p and q have the same truth value, and is false otherwise. Biconditional statements are also called bi-implications. Example: P - ‘I am going to town’ Q - ‘It is going to rain’ P Q: ‘I am going to town if and only if it is going to rain.’ 20 21 Translating English •Breaking assertions into component propositions look for the logical operators! •Example: ‘If I go to Harry’s or go to the country I will not go shopping.’ P: I go to Harry’s Q: I go to the country R: I will go shopping If......P......or.....Q.....then....not.....R (P Q) ¬ R 22 Constructing a truth table 1. 2. 3. 4. one column for each propositional variable one for the compound proposition count in binary n propositional variables = 2n rows • Construct the truth table for (P ¬ Q) (PQ) • HW: Construct the truth table for (P Q) ¬ R 23 24 t01_1_008.jpg 25 • What is the real meaning of ¬ PQ ? a) (¬ P) Q b) ¬ (PQ) • What is the real meaning of PQR ? a) (PQ)R b) P(QR) • What is the real meaning of P QR ? a) (P Q)R b) P (QR) 26 27 Logic and Bit Operations • Example 20 Find the bitwise OR, bitwise AND, and bitwise XOR of the bit strings 01 1011 0110 and 11 0001 1101. 28 Logic Puzzles • Example 18: There are two kind of inhabitants, knights, who always tell the truth, and their opposites, knaves, who always lie. You encounter two people A and B. What are A and B if A says “B is a knight” and B says “The two of us are opposite type”? 29 Terms • • • • • • Proposition Negation Conjection Disjunction Exclusive OR Implication • Inverse • Converse • Contrapositive 30