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Nested Quantifiers Section 1.4

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Recap Section 1.3 A predicate is generalization of a proposition. –It is a proposition that contains variables. A predicate becomes a proposition if the variable(s) contained is(are) –Assigned specific value(s) –Quantified Universe of discourse : the particular domain of the variable in a propositional function

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Recap Section 1.3 Universal quantification –P(x) is true for ALL the values of x in the universe of discourse. – x P(x). –Remember A ll. –for all x, P(x) If the elements in the universe of discourse can be listed, U = { x 1, x 2, …, x n } x P(x) P(x 1 ) P(x 2 ) … P(x n )

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Recap Section 1.3 Existential quantification – P(x) is true FOR SOME x in the universe of discourse, i.e. EXIST some x – x P(x) –Remember, Exist –for some x, P(x) If the elements in the universe of discourse can be listed, U = { x 1, x 2, …, x n } x P(x) P(x 1 ) P(x 2 ) … P(x n )

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Recap Section 1.3 Universal quantifiers usually take implications All CS students are smart students. x [C(x) S(x)] Existential quantifiers usually take conjunctions Some CS students are smart students. x [C(x) S(x)]

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Recap Section 1.3 Summary of quantifiers x P(x) –True when: P(x) is true for every x –False when: P(x) is false for at least one x. x P(x) –True when: P(x) is true for at least one x –False when: P(x) is false for every x Negation changes a universal to an existential and vice versa, and negates the predicate ~ x P(x) x ~P(x)

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Recap Section 1.3 Quick examples Z(13b) Determine truth value. U={Z} – n (2n = 3n) R(16b) Determine truth value U={R} – n (x 2 = -1) Exercise 17

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Nested Quantifiers Quantifiers that occur within the scope of other quantifiers Example: R P ( x, y ): x + y = 0, U={ R } x y P ( x, y )

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Quantifications of Two Variables For all pair x, y P ( x, y ). x y P ( x, y ) y x P ( x, y ) For every x there is a y such that P ( x, y ). x y P ( x, y ) There is an x such that P ( x, y ) for all y. x y P ( x, y ) There is a pair x, y such that P ( x, y ). x y P ( x, y ) y x P ( x, y )

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Translating statements with nested quantifiers U = {all real numbers} x y ( x + y = y + x ) x y ( x + y = 0) x y ( ( x > 0) ( y < 0) ( xy < 0) ) U = {all students in cs2813} C ( x ): x has a computer F ( x, y ): x and y are friends x ( C( x ) y ( C ( y ) F ( x, y )) )

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Translating Sentences U = {all people} –If a person is female and is a parent, then this person is someones mother. U = {all integers} –The sum of two positive integers is positive.

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Is the order of quantifiers important? If the quantifiers are of the same type, then order does not matter If the quantifiers are of different types, then order is important

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Example RU={R} Q(x,y): x+y=0 What are the truth values for y x Q(x,y) and x y Q(x,y) y x Q(x,y): There exist at least one y such that for every real number x, Q(x,y) is true, i.e. x+y=0. FALSE (not for every, only when y is –x). But… x y Q(x,y): For every real number x, there is a real number y such that Q(x,y) is true, i.e x+y =0. TRUE (for every x when y is –x)

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