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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 All. “for all x, P(x)” If the elements in the universe of discourse can be listed, U = {x1, x2, …, xn} x P(x) P(x1) P(x2) … P(xn)

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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 = {x1, x2, …, xn} x P(x) P(x1) P(x2) … P(xn)

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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) ~x P(x) x ~P(x)

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**Recap Section 1.3 Quick examples**

(13b) Determine truth value. U={Z} n (2n = 3n) (16b) Determine truth value U={R} n (x2 = -1) Exercise 17

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Nested Quantifiers Quantifiers that occur within the scope of other quantifiers Example: 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). xy P(x,y) yx P(x,y) For every x there is a y such that P(x,y). xy P(x,y) There is an x such that P(x,y) for all y. xy P(x,y) There is a pair x,y such that P(x,y). xy P(x,y) yx 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 someone’s 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 U={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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