Example every real number has an inverse is x yx y 0

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Example : “Every real number has an inverse” is x y(x + y = 0) where the domains of x and y are the real numbers. We can also think of nested propositional functions: x y(x + y = 0) can be viewed as x Q ( x ) where Q ( x ) is y P(x, y) where P(x, y) is (x + y = 0)
Thinking of Nested Quantification Nested Loops To see if x yP (x,y) is true, loop through the values of x : At each step, loop through the values for y . If for some pair of x and y, P(x,y) is false, then x yP(x,y) is false and both the outer and inner loop terminate. x y P(x,y) is true if the outer loop ends after stepping through each x . To see if x yP(x,y) is true, loop through the values of x : At each step, loop through the values for y. The inner loop ends when a pair x and y is found such that P ( x , y) is true. If no y is found such that P ( x , y) is true the outer loop terminates as x yP(x,y) has been shown to be false. x y P(x,y) is true if the outer loop ends after stepping through each x . If the domains of the variables are infinite, then this process can not actually be carried out.
Order of Quantifiers Examples : 1. Let P(x,y) be the statement “ x + y = y + x .” Assume that U is the real numbers. Then x yP(x,y) and y xP(x,y) have the same truth value. 2. Let Q(x,y) be the statement “ x + y = 0 .” Assume that U is the real numbers. Then x yP(x,y) is true , but y xP(x,y) is false.
Questions on Order of Quantifiers Example 1 : Let U be the real numbers, Define P(x,y) : x ∙ y = 0 What is the truth value of the following: 1. x yP(x,y) Answer: False 2. x yP(x,y) Answer: True 3. x y P(x,y) Answer: True 4. x y P(x,y) Answer: True
Questions on Order of Quantifiers Example 2 : Let U be the real numbers, Define P(x,y) : x / y = 1 What is the truth value of the following: 1. x yP(x,y) Answer: False 2. x yP(x,y) Answer: True 3. x y P(x,y) Answer: False 4. x y P(x,y) Answer: True
Quantifications of Two Variables Statement When True? When False P ( x , y ) is true for every pair x , y . There is a pair x, y for which P ( x,y ) is false. For every x there is a y for which P ( x,y ) is true. There is an x such that P ( x,y ) is false for every y . There is an x for which P ( x,y ) is true for every y . For every x there is a y for which P (x,y) is false. There is a pair x, y for which P ( x,y ) is true. P (x,y) is false for every pair x,y
Translating Nested Quantifiers into English Example 1 : Translate the statement x (C(x )∨ y (C(y ) ∧ F(x, y))) where C(x) is “ x has a computer,” and F ( x , y ) is “ x and y are friends,” and the domain for both x and y consists of all students in your school. Solution : Every student in your school has a computer or has a friend who has a computer.
Translating Mathematical Statements into Predicate Logic Example : Translate “The sum of two positive integers is always positive” into a logical expression.
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