Lecture 3

1.3 Predicates and Quantifiers 1.4 Nested Quantifiers

Predicates

The statement "x is greater than 3" has two parts. The first part, "x", is the subject of the statement. The second part, "is greater than 3" is called the predicate.

Quantifiers

Universal Quantification

Existential Quantification

Logical Equivalences Involving Quantifiers

Statements involving predicates and quantifiers are logically equivalent iff they have the same

truth value for all applications and for all domains of discourse.

)()()()( xQxxPxxQxPx

)()( xPxxxP

)()( xPxxxP

)()()()( xQxPxxQxPx

News Flash! Predicates are not propositions!they are propositional functions

subject predicate proposition

x x was a man. Daniel Boone was a man.

x x > 3 2 > 3

x,y Q(x,y) Q(2,3)

x,y,z R(x,y,z)=> x+y=z R(1,2,3) true

R(0,0,1) false

A predicate can be turned into a proposition by instantiation of its variables...

or by quantification...

subject predicate proposition

x x was a man

x x>3

x,y Q(x,y)

x,y,z R(x,y,z)=> x+y=z

)manawasx(x

)x(x 3

)y,x(Qyx

zyxzyx

Binding Variables

DeMorgan's Laws for Quantifiers

)x(P)x(P)x(P)x(xP n 21

Universal Quantifiers and Negation

For all x, P(x) is true.

To show that the universal quantifier is not true we need only to show a counterexample. That is we demonstrate that the negation of the universal quantifier is true.

)x(Px)x(xP

)x(P)x(P)x(P

)x(P)x(P)x(P

)x(Px)x(xP

n

n

21

21

Existential Quantifier and Negation

Existential quantifiers state that in the domain of discourse there exists as least one member for which the predicate is true.

)x(P)x(P)x(P)x(Px n 21

The negation of the existential quantifier says that none of the members of the domain of discourse make the predicate true.

)x(P)x(P)x(P

)x(P)x(P)x(P)x(Px

n

n

21

21

)x(B)x(F)x(Mx

)bat,x(T)x(F)mammal,x(Tx

There is more than one way to x a y.

We can express the statement,

"There is an animal that is a mammal and can fly but is not a bat", as a logical proposition,

where,

M(x) = x is a mammal F(x) = x can fly B(x) = x is a bat

Alternatively we can create predicates with multiple variables, such as

T(x,y) = returns true if x is a y

which allows us to rewrite the logical proposition as,

Understanding compound quantifiers

z)yx()zy(xzyx

xyyxyx

yxyx

0

For all x there exists a y such that x+y=0 - additive inverse

For all x and y, x+y is equal to y+x. - commutative law

For all x,y and z, x plus the quantity y+z is equal to the quantity x+y plus the value z. - associative law

Math Expressions

)y,x(F)y(Cy)x(Cx

Logical Expressions

C(x) - x owns a computerF(x,y) - x and y are friends

"Every student either owns a computer or has a friend that owns a computer."

Order of Quantifiers

Nested Quantifiers: An Example

Negating Compound Quantifiers... Au contraire, mon ami

Express the negation of the statement

so that no negation precedes a quantifier.

1

1

1

1

1

xyyx

xyyx

xyyx

xyyx

xyyx

It is not true that...

There exists some x such that there does not exist a y that makes xy=1

There exists some x such that for all y, x times y does not equal 1.

Compound Quantifiers as Nested Loops

Nested Loons

zyxzyx

isa_z = false;for x in x1..xn loop for y in y1..yn loop for z in z1..zn loop if x+y=z then isa_z = true; end if; end loop; end loop;end loop;

Propositional quantifiers imply an evaluation of the predicate for all possible values in the domain of discourse. Sometimes the domain is infinite and possibly uncountable, but we can still consider the comparison as being implemented as a nested loop.

In the code segment above, is there a more efficient way to represent existential quantifiers?