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Lec04_FirstOrderLogic__part_1_

# Lec04_FirstOrderLogic__part_1_ - TDS1191 Discrete...

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[Lecture04][First Order Logic (part 1)] Discrete Structures 1 TDS1191 Discrete Structures Lecture 04 First Order Logic (part 1) Multimedia University Trimester 2, Session 2011/2012

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[Lecture04][First Order Logic (part 1)] Discrete Structures 2 Statements about infinitely many objects, Statements about infinitely many objects, quantification statements like quantification statements like some of them … some of them … at least one of them … at least one of them … all of them … all of them … are difficult to express in propositional logic. are difficult to express in propositional logic. Example 1: “Every student in Discrete Structures class passed the final exam” is true, but no rules of propositional logic allows us to conclude the truth of “Amy passed her final exam of Discrete Structures” given that Amy is a student in the Discrete Structures class. Why we need first order logic? Example 2: Assuming that we are designing a computer specification. Given an input x > 10, the system should generate an error message. Without any value x , this statement is not a proposition. But what if we assign a value to x ? Let P(x) denotes the statement “ x > 10”. What are the truth values of the propositions P(3) and P(15) ?
[Lecture04][First Order Logic (part 1)] First Order Logic = First Order Predicate Logic = First Order Predicate Calculus 3

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[Lecture04][First Order Logic (part 1)] Discrete Structures 4 Predicate Logic A complete statement contains: a predicate with variable(s). Variable is what/whom the statement is about. Predicate tells something about the subject. e.g. Dog barks; (Dog = Variable, barks = Predicate) Refer to Example 2, P(x) , P represents the predicate, x > 10”, x is the variable. A complete statement contains: a predicate with variable(s). Variable is what/whom the statement is about. Predicate tells something about the subject. e.g. Dog barks; (Dog = Variable, barks = Predicate) Refer to Example 2, P(x) , P represents the predicate, x > 10”, x is the variable.
[Lecture04][First Order Logic (part 1)] Discrete Structures 5 Predicate Logic Predicate can be used to express the meaning of wide range of statements in mathematics and computer science in ways that permit us to reason and explore relationships between objects . Predicate can be used to express the meaning of wide range of statements in mathematics and computer science in ways that permit us to reason and explore relationships between objects . Sometimes, we may have 3 variables for a predicate: Q(x,y,z) denotes the statement “ x = y+z+30 ”. What are the truth values of the propositions Q(15,10,20) and Q(54,14,10) ?

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[Lecture04][First Order Logic (part 1)] Discrete Structures 6 Quantifiers Quantifiers are involved when we wish to express the predicate to be true under certain range of elements.
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Lec04_FirstOrderLogic__part_1_ - TDS1191 Discrete...

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