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Published on September 13, 2007

Author: AscotEdu

Source: authorstream.com

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Quantifiers:  Quantifiers This presentation is prepared by Yufei Tao. http://www.cse.cuhk.edu.hk/~taoyf Limitation of the operators discussed:  Limitation of the operators discussed So far we cannot model the following propositions: Every student is smart. At least one egg is bad in this basket. Everybody has parents. No person is allowed beyond this door. … The common thing in the above propositions is that they concern quantities. Propositional function:  Propositional function P(x): x is an even number. This function itself does not have a true or false value. It does, however, once x is assigned. We say that P is a predicate, and x a variable. Example: P(2) is a proposition whose value is true. P(3) is a false proposition. Propositional function (cont.):  Propositional function (cont.) May contain multiple variables. P(x, y): x + y = 5. P(2, 3) = true P(2, 4) = false Universe:  Universe I.e., domain of a variable. P(x): x is an even number. We also need to specify the universe explicitly to complete the definition. Example universes: The set of all integers All integers below 50 … Universal quantifier:  Universal quantifier x P(x) In English: For all x, P(x) holds. It is a proposition. P(x): x is an even number. Universe of x = {all integers}. x P(x) = false. P(x): x andlt; x + 1. Universe of x = {all real numbers}. x P(x) = true. P(x) = x has parents. Universe of x = {all human beings} x P(x) = true. Universal quantifier (cont.):  Universal quantifier (cont.) x P(x) When the universe of x can be enumerated as {x1, x2, …, xn}, we have x P(x) = P(x1)  P(x2)  …  P(xn) P(x): x has parents. Universe of x = {you and me} x P(x) = P(you)  P(me). Universal quantifier (cont.):  Universal quantifier (cont.) x y P(x, y) P(x, y) = x + y andgt; 2(x  y)1/2. Universe of x = Universe of y = {All real numbers} x y P(x, y) = false. It would be true if ‘andgt;’ was replaced with ‘’. Existential quantifier:  Existential quantifier x P(x) There exists an x that satisfies P(x). P(x): x is an even number. Universe of x = {all integers}. x P(x) = true. P(x): x andlt; x + 1. Universe of x = {all real numbers}. x P(x) = true. P(x) = x has no parents. Universe of x = {all human beings} x P(x) = false. Existential quantifier (cont.):  Existential quantifier (cont.) x P(x) When the universe of x can be enumerated as {x1, x2, …, xn}, we have x P(x) = P(x1)  P(x2)  …  P(xn) P(x): x is an even number. Universe of x = {1, 2, 3} x P(x) = P(1)  P(2)  P(3). Mixing the two types of quantifiers:  Mixing the two types of quantifiers P(x, y): Man x loves woman y. Universe of x = {all men}. Universe of y = {all women} x y P(x, y): Every man loves all the women in the world. False x y P(x, y): Every man loves a women. True Bounded/free variables:  Bounded/free variables x y (P(x, y)  Q(x))  R(z) x and y are bounded. z is free. We are interested only in the propositional functions where all the variables are bounded. Negation rule 1:  Negation rule 1 x P(x)  x P(x) P(x): x scores A in CSC3190. x P(x): Not every student scores A in CSC3190. x P(x): There exists a student that does not score A in CSC3190. Negation rule 1 (cont.):  Negation rule 1 (cont.) x P(x)  x P(x) This rule is easier to perceive when the universe of x can be enumerated as x1, x2, …, xn. We know: x P(x)  P(x1)  P(x2)  …  P(xn) x P(x)  P(x1)  P(x2)  …  P(xn) Hence: x P(x)  (P(x1)  P(x2)  …  P(xn))  P(x1)  P(x2)  …  P(xn)  x P(x) Negation rule 2:  Negation rule 2  x P(x)  x P(x) P(x): x passes CSC3190.  x P(x): There does not exist any student who passes CSC3190. x P(x): All students fail in CSC3190. When the universe of x can be enumerated, the original equation can be established with the distributive law in the same way as negation rule 1. Interesting translation 1:  Interesting translation 1 Some creatures are lions, and some lions do not drink coffee. Universe of x = {all creatures} P(x): x is a lion. Q(x): x drinks coffee. x (P(x)  Q(x)) How about x (P(x)  Q(x))? This proposition is true as long as some creatures are not lions. Interesting translation 2:  Interesting translation 2 No large birds live on honey. Universe of x = {all birds} P(x) = x is large. Q(x) = x lives on honey. x (P(x)  Q(x)) or x (P(x)  Q(x)) Interesting translation 3:  Interesting translation 3 Birds that do not live on honey are dull in color. Universe of x = {all birds} P(x) = x lives on honey. Q(x) = x is dull in color. x (P(x)  Q(x)) How about x (P(x)  Q(x))? This proposition evaluates to false for birds that actually live on honey. The original English sentence does not have such implication.

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