DM Lecture 2.pdf - DISCRETE STRUCTUERS COURSE I NSTRUCTOR MUHAMMAD SAIF UL I SLAM Course Outline Logic and Proofs(Chapter 1 Sets and Functions Relations

DM Lecture 2.pdf - DISCRETE STRUCTUERS COURSE I NSTRUCTOR...

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DISCRETE STRUCTUERS COURSE INSTRUCTOR: MUHAMMAD SAIF UL ISLAM
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© Saif ul Islam - 2017 Course Outline Logic and Proofs (Chapter 1) Sets and Functions Relations Number Theory Combinatorics and Recurrence Graphs Trees Discrete Probability FAST NUCES DISCRETE STRUCTURES 2
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© Saif ul Islam - 2017 Lecture Outline Propositional Logic Propositional Operators Conditional Statements Compound Propositions and Truth Tables Precedence of Logical Operators Applications of Propositional Logic Propositional Equivalences FAST NUCES DISCRETE STRUCTURES 3
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© Saif ul Islam - 2017 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 .” Examples: 1. Washington, D.C., is the capital of the United States of America. 2. Toronto is the capital of Canada. 3. 1 + 1 = 2. 4. 2 + 2 = 3. 5. What time is it? 6. Read this carefully. 7. x + 1 = 2. FAST NUCES DISCRETE STRUCTURES 4 * Declarative sentences: They don't ask questions, make commands, or make statements with emotion. Identify True, False and non-propositions 1,3 are True 5,6,7 are not propositions Rest are False
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© Saif ul Islam - 2017 Propositional Logic Some More Examples: 1. He is a college student. 2. x+4 > 9 3. It will rain tomorrow in Karachi. 4. Blue is the best color to paint a house. 5. The integer n is even. FAST NUCES DISCRETE STRUCTURES 5
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© Saif ul Islam - 2017 Propositional Operators FAST NUCES DISCRETE STRUCTURES 6
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© Saif ul Islam - 2017 Propositional Operators 1. Negation Let p be a proposition. The negation of p , denoted by p is the statement “It is not the case that p The proposition p is read “not p .” The truth value of the negation of p , p , is the opposite of the truth value of p . * Other notations you might see are p , − p , p ′, N p , and ! p . Example: p = “Michael’s PC runs Linux p = “It is not the case that Michael’s PC runs Linux.” OR p = “Michael’s PC does not run Linux.” FAST NUCES DISCRETE STRUCTURES 7
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© Saif ul Islam - 2017 Propositional Operators 2. Conjunction Let p and q be propositions. The conjunction of p and q , denoted by p q , is the proposition p and q .” The conjunction p q is true when both p and q are true and is false otherwise. Example: Rebecca’s PC has more than 16 GB free hard disk space, and the processor in Rebecca’s PC runs faster than 1 GHz. p = “Rebecca’s PC has more than 16 GB free hard disk space.” q = “The processor in Rebecca’s PC runs faster than 1 GHz.” p q FAST NUCES DISCRETE STRUCTURES 8
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© Saif ul Islam - 2017 Propositional Operators FAST NUCES DISCRETE STRUCTURES 9
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© Saif ul Islam - 2017 Propositional Operators 3. Disjunction Let p and q be propositions. The disjunction of p and q
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