Sections 1.3 and 1.4

Sections 1.3 and 1.4 Predicates & Quantifiers

Propositional Functions • In a mathematical assertion, such as x < 3, there are two parts: • the subject, which is the variable (x in this case) • the predicate, which is the property the subject may (or may not) have (< 3 in the example) • We can denote such a predicate as a propositional function on x, or P(x) • When x has a value, P(x) is a proposition

Propositional Functions A propositional function can include more than one variable; for example: Let Q(x,y) denote the statement “x is the capital of y” What are the truth values of: Q(Denver, Colorado) Q(Detroit, Michigan) Q(Massachusetts, Boston) Q(New York, New York)

Propositional Functions and Programs • In a program, a selection structure involves evaluating a propositional function • For example: • if (x > 0) • x++; Means let P(x) = x > 0 if P(x) is true, increment x otherwise, do nothing

Quantifiers • As we have seen, we can create a proposition from a propositional function by assigning a value to its subject(s) • We can also create a proposition by quantifying the propositional function • There are two types of quantifiers: • universal • existential

Universal Quantification • The domain of a problem is called its universe of discourse • The universal quantification of P(x) is the assertion that P(x) is true for all values of x in the universe of discourse • Universal quantification is denoted xP(x) which may be read “for all (or every) value of x, P(x) is true”

Universal Quantification • Suppose P(x) is the statement “x spends more than 3 hours every day in class” and the universe of discourse is the set of all students • Then xP(x) would be the statement “All students spend more than 3 hours every day in class” • This could also be expressed: x(S(x)  P(x)) where S(x) = “x is a student”

Universal Quantification & Conjunction When all elements in a universe of discourse can be listed, e.g. (x1, x2, … , xn) then xP(x) can be written as: P(x1)  P(x2)  …  P(xn) For example, suppose P(x) = x < x2 where the universe of discourse is the positive integers less than 5 For xP(x) to be true, P(0)  P(1)  P(2)  P(3)  P(4) would have to be true; since 0 < 0 and 1 < 1 are untrue, xP(x) is false

Existential Quantification • The assertion that there is an element with a certain property • The notation xP(x) means there exists at least one element x in the universe of discourse for which P(x) is true

Existential Quantification & Disjunction When all elements in a universe of discourse can be listed, e.g. (x1, x2, … , xn) then xP(x) can be written as: P(x1)  P(x2)  …  P(xn) For example, suppose P(x) = x < x2 where the universe of discourse is the positive integers less than 5 For xP(x) to be true, P(0)  P(1)  P(2)  P(3)  P(4) would have to be true; since 2 < 4, xP(x) is true

Universal vs. Existential Quantification xP(x) True if P(x) is true False if there for all values of x is at least one in universe of value x where discourse P(x) is false xP(x) True if there exists False if P(x) is at least one value x false for every x for which P(x) in the universe is true of discourse

Quantifiers & Looping • Can be helpful to think in terms of looping and searching when seeking truth value of a quantification • For xP(x), loop through all N values in universe of discourse; if all are true, xP(x) is true • For xP(x), loop through all N values; if any are true, xP(x) is true

Binding Variables • A variable is said to be bound if it has a value assigned to it or if a quantifier is used on it • A variable is free if neither of these conditions applies

Binding Variables • In order to turn a propositional function into a proposition, all variables in the function must be bound • Can have multiple quantifications for propositional functions involving more than one variable • The order of quantifiers is important unless all are of the same type (all existential or all universal)

Example Are these quantifications logically equivalent? x y P(x,y) y x P(x,y) x y P(x,y) y x P(x,y) There exists an x for For every value of y there which P(x,y) is true is a value of x for which for every value of y P(x,y) is true So there must be an x So x can depend on y for which the function is true regardless of y - x is an independent constant If the first is true, the second is true - but not vice-versa

Translating Quantified Expressions Suppose W(x,y) is the propositional function “student x has taken class y” and the universe of discourse for x is students in this class, while the universe of discourse for y is classes at Kirkwood Then the quantification y(W(Tim,y)  W(Dan,y)) means there exists a class at Kirkwood for which the statements “Tim has taken this class” and “Dan has taken this class” is true What does yx(W(x,y)) mean?

Translating Quantified Statements Suppose the universe of discourse for x, y and z is the set of real numbers. What is the meaning of the following quantification? x y z (z * (x + y) = (z * x) + (z * y)) This is the distributive law for multiplication: For all real numbers x, y and z, the product of z and the sum of x and y is equal to the sum of the products of z and x and z and y

Translating Quantified Statements Can translate from English into logical expressions, as well as vice versa For example, suppose L(x,y) is the propositional function x loves y, and the universe of discourse for x and y is all the people in the world The quantification “Everybody loves me” can be denoted as: x L(x, me) How would you denote “I love everybody” and “Somebody loves me?”

More “love” examples There is somebody whom everybody loves: y x L(x,y) Everybody loves somebody: x y L(x,y) There is somebody whom nobody loves: y x (L(x,y))

Nested Loops & Multiple Quantifications • x y P(x,y): loop through all values of x; in each x loop, loop through all values of y; true if no false values found • x y P(x,y): loop through all values of x; loop through all y values at each x until a y is found for which P(x,y) is true - proposition is true if there is such a y for every x

Nested Loops & Multiple Quantifications • x y P(x,y): loop through all values of x until we find an x for which P(x,y) is true for all values of y • x y P(x,y): start looping through x values; at each x, loop through y values - if we hit at least one x,y where P(x,y) is true, then the proposition is true

Summary of truth values for multiple quantifications x y P(x,y) True if P(x,y) is False if there is at least one y x P(x,y) true for every x,y pair for which P(x,y) x,y pair is false x y P(x,y) True if there is a False if there is an x for y that makes P(x,y) which P(x,y) is false for true for every x every y x y P(x,y) True if there is an False if for every x there x for which P(x,y) is a y for which P(x,y) true for every y is false x y P(x,y) True if there is at False if P(x,y) is false y x P(x,y) least 1 x,y pair for for every x,y pair which P(x,y) is true

Negations of quantified expressions • The negation of a universal quantification is the existential quantification of the negation • The negation of an existential quantification is the universal quantification of the negation

Negations of quantified expressions x P(x)  x P(x) True if P(x) is false False if there is an x for for every value of x which P(x) is true x P(x)  x P(x) True if there exists False if P(x) is true for an x for which P(x) every x is false

Section 1.3 Predicates & Quantifiers - ends -

Sections 1.3 and 1.4