DM Lecture 3.pdf - DISCRETE STRUCTUERS COURSE I NSTRUCTOR MUHAMMAD SAIF UL I SLAM Course Outline Logic and Proofs(Chapter 1 Sets and Functions Relations

# DM Lecture 3.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
© Saif ul Islam - 2017 Lecture Outline Limitations of Propositional Logic Predicates Quantifiers Logical Equivalences Involving Quantifiers Translating from English to Logic Examples Nested Quantifiers Order of Quantifiers Translating Nested Quantifiers FAST NUCES DISCRETE STRUCTURES 3

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© Saif ul Islam - 2017 Limitations of Propositional Logic Propositional logic cannot adequately express the meaning of all statements in mathematics and in natural language Example: If we want to say “ All students are present.” OR “ Some students are present ”, We need N number of statement to represent it. Where N refers to the number of students . Can’t be represented in propositional logic. Need a language that talks about objects, their properties, and their relations. FAST NUCES DISCRETE STRUCTURES 4
© Saif ul Islam - 2017 Predicates Predicate logic can be used to express the meaning of a wide range of statements in mathematics and computer science in ways that permit us to reason and explore relationships between objects Predicate logic uses the following new features: Variables: x , y , z Predicates: P ( x ), M ( x ) Quantifiers ( to be covered in a few slides ): Propositional functions are a generalization of propositions. They contain variables and a predicate, e.g., P ( x ) Variables can be replaced by elements from their domain . FAST NUCES DISCRETE STRUCTURES 5

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© Saif ul Islam - 2017 Predicates Example: The statement x is greater than 3” has two parts. 1- The variable x , is the subject of the statement. 2- The predicate , “is greater than 3” refers to a property that the subject of the statement can have. We can denote the statement “ x is greater than 3” by P ( x ), where P denotes the predicate “is greater than 3” and x is the variable. FAST NUCES DISCRETE STRUCTURES 6
© Saif ul Islam - 2017 Propositional Functions Once a value has been assigned to the variable x , the statement P ( x ) becomes a proposition and has a truth value Example: Let P ( x ) denote the statement “ x > 3 .” What are the truth values of P (4) and P (2)? We obtain the statement P (4) by setting x = 4 in the statement “ x > 3 .” P (4 ), which is the statement “4 > 3,” is true . P (2 ), which is the statement “2 > 3 ,” is false . FAST NUCES DISCRETE STRUCTURES 7

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© Saif ul Islam - 2017 Propositional Functions Example: Let Q ( x, y ) denote the statement “ x = y + 3 .” What are the truth values of the propositions Q (1 , 2) and Q (3 , 0)?
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