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Sections 1.3 and 1.4

Predicates & Quantifiers

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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

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Propositional Functions

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)

A propositional function can include more than one variable; forexample:

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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 > 0if P(x) is true, increment xotherwise, do nothing

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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

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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”

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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”

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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 ofdiscourse is the positive integers less than 5

For xP(x) to be true, P(0) P(1) P(2) P(3) P(4) wouldhave to be true; since 0 < 0 and 1 < 1 are untrue, xP(x) is false

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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

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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 ofdiscourse is the positive integers less than 5

For xP(x) to be true, P(0) P(1) P(2) P(3) P(4) wouldhave to be true; since 2 < 4, xP(x) is true

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Universal vs. Existential Quantification

xP(x) True if P(x) is true False if therefor all values of x is at least onein universe of value x wherediscourse P(x) is false

xP(x) True if there exists False if P(x) isat least one value x false for every xfor which P(x) in the universeis true of discourse

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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

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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

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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)

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ExampleAre 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 therewhich P(x,y) is true is a value of x for whichfor every value of y P(x,y) is true

So there must be an x So x can depend on yfor which the function istrue regardless of y - x isan independent constant

If the first is true, the second is true - but not vice-versa

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Translating Quantified Expressions

Suppose W(x,y) is the propositional function “student x has takenclass y” and the universe of discourse for x is students in thisclass, while the universe of discourse for y is classes at Kirkwood

Then the quantification y(W(Tim,y) W(Dan,y)) meansthere exists a class at Kirkwood for which the statements“Tim has taken this class” and “Dan has taken this class” is true

What does yx(W(x,y)) mean?

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Translating Quantified Statements

Suppose the universe of discourse for x, y and z is the set ofreal 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 ofx and y is equal to the sum of the products of z and x and z and y

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Translating Quantified Statements

Can translate from English into logical expressions, as well asvice versa

For example, suppose L(x,y) is the propositional functionx loves y, and the universe of discourse for x and y is all thepeople in the world

The quantification “Everybody loves me” can be denoted as:x L(x, me)

How would you denote “I love everybody” and “Somebodyloves me?”

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More “love” examplesThere 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))

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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

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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

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Summary of truth values for multiple quantifications

x y P(x,y) True if P(x,y) is False if there is at least oney x P(x,y) true for every x,y pair for which P(x,y)

x,y pair is falsex 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 fortrue for every x every y

x y P(x,y) True if there is an False if for every x therex 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 falsey x P(x,y) least 1 x,y pair for for every x,y pair

which P(x,y) is true

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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

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Negations of quantified expressions

x P(x) x P(x)True if P(x) is false False if there is an x forfor 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 foran x for which P(x) every xis false

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Section 1.3

Predicates & Quantifiers

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