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ICS 253: Discrete Structures I Dr. Nasir Al-Darwish Computer Science Department King Fahd University of Petroleum and Minerals Spring Semester 2014 (2013-2) Predicates and Quantifiers

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KFUPM: Dr. Al-Darwish © Propositional Predicate Definition: A propositional predicate P(x) is a statement that has a variable x. Examples of P(x) P(x) = “The Course x is difficult” P(x) = “x+2 < 5” Note: a propositional predicate is not a proposition because it depends on the value of x. Example: If P(x) = “x > 3”, then P(4) is true but not P(1). A propositional predicate is also called a propositional function

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KFUPM: Dr. Al-Darwish © Propositional Predicate – cont. It also possible to have more than one variable in one predicate, e.g., Q(x,y) =“x > y-2” P(x) is a function (a mapping) that takes a value for x and produce either true or false. Example: P(x) = “x 2 > 2”, P: Some Domain {T, F} Domain of x is called the domain (universe) of discourse

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KFUPM: Dr. Al-Darwish © Quantification A predicate (propositional function) could be made a proposition by either assigning values to the variables or by quantification. Predicate Calculus: Is the area of logic concerned with predicates and quantifiers.

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KFUPM: Dr. Al-Darwish © Quantifiers 1. Universal quantifier: P(x) is true for all (every) x in the domain. We write x P(x) 2. Existential quantifier: there exists at least one x in the domain such that P(x) is true. We write x P(x) 3. Others: there exists a unique x such that P(x) is true. We write !x P(x)

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KFUPM: Dr. Al-Darwish © Universal Quantification Uses the universal quantifier (for all) x P(x) corresponds to “p(x) is true for all values of x (in some domain)” Read it as “for all x p(x)” or “for every x p(x)” Other expressions include “for each”, “all of”, “for arbitrary”, and “for any” (avoid this!) A statement x P(x) is false if and only if p(x) is not always true (i.e., P(x) is false for at least one value of x) An element for which p(x) is false is called a counterexample of x P(x); one counterexample is all we need to establish that x P(x) is false

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KFUPM: Dr. Al-Darwish © Universal Quantification - Examples Example 1: Let P(x) be the statement “x + 1 > x”. What is the truth value of ∀ x P(x), where the domain for x consists of all real numbers? Solution: Because P(x) is true for all real numbers x, the universal quantification ∀ x P(x) is true. Example 2: Suppose that P(x) is “x 2 > 0 ”. What is the truth value of ∀ x P(x), where the domain consists of all integers. Solution: We show ∀ x P(x) is false by a counterexample. We see that x = 0 is a counterexample because for x = 0, x 2 = 0, thus there is some integer x for which P(x) is false.

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KFUPM: Dr. Al-Darwish © Existential Quantification Uses the existential quantifier (there exists) x P(x) corresponds to “There exists an element x (in some domain) such that p(x) is true” In English, “there is”, “for at least one”, or “for some” Read as “There is an x such that p(x)”, “There is at least one x such that p(x)”, or “For some x, p(x)” A statement x P(x) is false if and only if “for all x, P(x) is false”

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KFUPM: Dr. Al-Darwish © Existential Quantification - Examples Example 1: Let P(x) denote the statement “x > 3”. What is the truth value of ∃ x P(x), where the domain for x consists of all real numbers? Solution: Because “x > 3” is true for some values of x, for example, x = 4, the existential quantification ∃ x P(x) is true. Example 2: Let Q(x) denote the statement “x = x + 1”. What is the truth value of ∃ x Q(x), where the domain consists of all real numbers? Solution: Because Q(x) is false for every real number x, the existential quantification ∃ x Q(x) is false.

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KFUPM: Dr. Al-Darwish © Predicates and Negations PredicateWhen trueWhen falseNegation x P(x)x P(x) For all x, P(x) is trueThere is at least one x s.t. P(x) is false x P(x)x P(x) x P(x)x P(x) There is at least one x s.t. P(x) is true For all x, P(x) is false x P(x)x P(x)

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KFUPM: Dr. Al-Darwish © Domain (or Universe) of Discourse Cannot tell if a quantified predicate P(x) is true (or false) if the domain of x is not known. The meaning of the quantified P(x) changes when we change the domain. The domain must always be specified when universal or existential quantifiers are used; otherwise, the statement is ambiguous.

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KFUPM: Dr. Al-Darwish © Quantification Examples P(x) = “x+1 = 2” Domain is R (set of real numbers) Proposition Truth Value x P(x) x P(x) x P(x) x P(x) !x P(x) !x P(x) F F T T T F

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KFUPM: Dr. Al-Darwish © Quantification Examples P(x) = “x 2 > 0” Domain Proposition Truth Value R x P(x) Z x P(x) Z - {0} x P(x) Z !x P(x) N={1,2,..} x P(x) F F T T F

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KFUPM: Dr. Al-Darwish © Quantification Examples Proposition Truth Value x R (x 2 x) !x R (x 2 < x) x (0,1) (x 2 < x) x {0,1} (x 2 = x) x P(x) F T F T T

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KFUPM: Dr. Al-Darwish © Logically Equivalence Definition: Two statements involving predicates & quantifiers are logically equivalent if and only if they have the same truth values independent of the domains and the predicates. Examples: ( x P(x) ) x P(x) ( x P(x) ) x P(x)

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KFUPM: Dr. Al-Darwish © Theorem If the domain of discourse is finite, say Domain = {x 1, x 2, …, x n }, then x P(x) x P(x) P(x 1 ) P(x 2 ) ... P(x n ) P(x 1 ) P(x 2 ) ... P(x n )

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KFUPM: Dr. Al-Darwish © Precedence , , , , , Example: ( ( x P(x) ) ( ( ! x Q(x) ) ( x P(x) ) ) )

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KFUPM: Dr. Al-Darwish © Correct Equivalences x ( P(x) Q(x) ) x P(x) x Q(x) This says that we can distribute a universal quantifier over a conjunction x ( P(x) Q(x) ) x P(x) x Q(x) This says that we can distribute an existential quantifier over a disjunction The preceding equivalences can be easily proven if we assume a finite domain for x = {x 1, x 2, …, x n } Note: we cannot distribute a universal quantifier over a disjunction, nor can we distribute an existential quantifier over a conjunction.

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KFUPM: Dr. Al-Darwish © Wrong Equivalences x ( P(x) Q(x) ) x P(x) x Q(x) Read as “there exists an x for which both P(x) and Q(x) are true is equivalent to there exists an x for which P(x) is true and there exists an x for which Q(x) is true”. One can construct an example that makes the above equivalence false. Consider the following x ( P(x) Q(x) ) = F but x P(x) x Q(x) = T, since P(1) is true and Q(2) is true x 12 P(x)P(x) TF Q(x)Q(x) FT

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KFUPM: Dr. Al-Darwish © Wrong Equivalences x ( P(x) Q(x) ) x P(x) x Q(x) One can construct an example that makes the above equivalence false. Consider the following x ( P(x) Q(x) ) = T but x P(x) x Q(x) = F F = F x 12 P(x)P(x) TF Q(x)Q(x) FT

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KFUPM: Dr. Al-Darwish © Wrong Equivalences ( x P(x) ) Q(x) x ( P(x) Q(x) ) Notice that the LHS = ( x P(x) ) Q(x) is not fully quantified. So it cannot be equivalent to RHS.

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KFUPM: Dr. Al-Darwish © Quantifiers with restricted domains What do the following statements mean for the domain of real numbers? Be careful about → and ˄ in these statements

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