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Lec01_Propositional_Logic

# Lec01_Propositional_Logic - TDS1191 Mark Distributions...

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TDS1191 Mark Distributions Tests (30%) Test 1, Week 6 (23 Nov, 2011), Lecture 1 – Lecture 5 Test 2, Week 10 (21 Dec, 2012), Lecture 6 – Lecture 10 Quizzes (10%) Week 3, 5, 7, 9, 11, 13 Homework (10%) Final Exam (50%) Text Book: Kenneth Rosen, Discrete Mathematics and Its Applications , 6/e, 2006, McGraw-Hill, ISBN: 0072880082 Discrete Structures 1

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Discrete Structures 2 TDS1191 Discrete Structures Lecture 01 Propositional Logic Multimedia University Trimester 2, Session 2011/2012 [Lecture01][Propositional Logic]
Discrete Structures 3 Logic and IT Logic is used to formalize (express) statements verify (prove) statements/computer programs (automatically) deduce new statements design computer circuits Constructing a mathematical (logical) proof is isomorphic with writing a computer program. [Daly, Waldron] [Lecture01][Propositional Logic]

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Discrete Structures 4 Proposition (Statement) : A proposition is a declarative sentence that is either true or false , but not both . Examples : 1. Please sit down. 2. x+1=2. 3. What time is it? Examples : 1. Please sit down. 2. x+1=2. 3. What time is it? [Lecture01][Propositional Logic] Examples : 1. Kuala Lumpur is the capital of Malaysia. 2. Peter is a boy. 3. 1+5=9. 4. Today is Tuesday. Examples : 1. Kuala Lumpur is the capital of Malaysia. 2. Peter is a boy. 3. 1+5=9. 4. Today is Tuesday. Propositions Not propositions
Discrete Structures 5 For a proposition p, we let [[p]] = truth value of the proposition p, i.e., [[p]]=T (true) if the proposition p is true. [[p]]=F (false) if the proposition p is false. Convention: We also use 1 for T and 0 for F. [Lecture01][Propositional Logic] Examples : 1. [[Kuala Lumpur is the capital of Malaysia]]=T 2. [[1+6=9]]=F 3. [[Today is Tuesday]]=? 4. [[Peter is a boy]]=? Examples : 1. [[Kuala Lumpur is the capital of Malaysia]]=T 2. [[1+6=9]]=F 3. [[Today is Tuesday]]=? 4. [[Peter is a boy]]=?

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Discrete Structures 6 Compound Propositions : If p and q are propositions, we can form new propositions from p and q using the logical operators (connectives): ¬ (negation), (conjunction), (disjunction), (exclusive or), (conditional), (bi-conditional) . 1. ¬p : “not p”, “it is not the case that p” 2. p q : “p and q” 3. p q : “p or q” 4. p q : “exclusive or of p and q” 5. p q : “p implies q”, “if p, then q”, “p only if q”, “p is sufficient for q”, “q is necessary for p”, “q if p”, “q when p”, “q whenever p”. 6. p q : “p if and only if q”, “p is necessary and sufficient for q”. [Lecture01][Propositional Logic] Truth Table: A table that displays the relationships between the truth values of propositions.
Discrete Structures 7 The truth table for logical operators ¬, , , , and : p q ¬p p q p q p q p q p q T T F T T F T T T F F F T T F F F T T F T T T F F F T F F F T T [Lecture01][Propositional Logic] Example 1 : p: It is sunny.

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