Nested loop
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The document discusses predicate logic and quantifiers. It defines predicate logic as using variables, quantifiers, and predicates to make statements about subjects. It explains common quantifiers like "for all" and "there exists" and how they are used. It also discusses how to translate statements with nested quantifiers and how to determine if quantified statements are true or false.
Nested loop
- 1. Nepal College of Informaiton Technology Balkumari, Lalitpur May 18, 2010 ॐ NEPAL COLLEGE OF INFORMATION TECHNOLOGY _ _ _ _ _ _ _ _ _ _ _ _ 2001
- 2. make statements about individual subjects predicate P(x) has two parts: variable ‘x’ is the subject of statement propositional function P is the property that the subject can have property of an inference May 18, 2010
- 3. property of description of subject in “domain or universe of discourse” P(x) predicate two condition: all values of ‘x’ true some values of ‘x’ true May 18, 2010 ‘ n’ objects ‘ x’ values
- 4. dynamic in nature logic operator are used so called predicate logic quantifiers & variables are used & variables bound the quantifier (universal or existential or both) i.e. symbolic logic May 18, 2010
- 5. P(x) : “x is man” For all values of ‘x’, if P(x) is true then x P(x) exists For some values of ‘x’, if P(x) is true then x P(x) exists “ All students in this class love discrete structure.” x love(x, discrete structure) “ All student love some subject.” x y love(x, y) May 18, 2010
- 6. Two quantifiers are nested if one is within the scope of the other, such as ∀ x ∃y (x + y = 0) Everything within the scope of a quantifier can be thought of as a propositional function. Define the propositional functions Q(x) : ∃y P(x, y) P(x, y) : x + y = 0 Then, we have ∀ x ∃y (x + y = 0) ≡ ∀x ∃y P(x, y) ≡ ∀x Q(x) . May 18, 2010
- 7. Translation of nested quantifiers can be done by: write out what the quantifiers and predicates in the expression means convert this meaning into a simpler sentence without using any of the variables Statements involving nested quantifiers can be negated by applying the rules for negating statements involving a single quantifier Table : Negating Quantifiers May 18, 2010 Negation Equivalent Statement When Is Negation True? When False? x P ( x ) x P ( x ) x P ( x ) x P ( x ) For every x , P ( x ) is false. There is an x for which P ( x ) is false. There is an x for which P ( x ) is true. P ( x ) is true for every x .
- 8. For example, to evaluate x y P(x, y) we loop through all the values of x , and for each x we loop through all the values of y. Table : Quantifications of Two Variables May 18, 2010 Statement When True? When False? x y P(x, y) y x P(x, y) P(x, y) is true for every pair x, y. There is a pair x, y for which P(x, y) is false. x y P(x, y) 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 all y. x y P(x, y) There is an x for which P(x, y) is true for all y. For every x there is a y for which P(x, y) is false. x y P(x, y) y x P(x, y) There is a pair x, y for which P(x, y) is true. P(x, y) is false for every pair x, y.
- 9. Example: Translate the statement “The sum of two positive integers is always positive” into a logical expression. Solution: May 18, 2010
- 10. K. Rosen, “ Discrete Mathematical Structures with Applications to Computer Science, WCB/ Mcgraw Hill ”, edition 6 2006 http://www.slideshare.net/../predicate-logic www.cs.odu.edu/~toida/nerzic/conten .. www.earlham.edu/../terms3.htm May 18, 2010
- 11. If you have any suggestion then let me know. May 18, 2010