Predicates and Quantified Statements

M260 3.1, 3.2

Predicate Example

• James is a student at Southwestern College.

• P(x,y)

• x is a student at y.

Predicate Truth Set

• P(x) is a predicate and x has domain D.

• The truth set of P(x):

• {x D P(x)}

• For all

• For every

• For arbitrary

• For any

• For each

• Given any

Universal Statement

x D, Q(x)

• When true

• When false

• Counter examples

• There exists

• We can find a

• There is at least one

• For some

• For at least one

Existential Statement

x D such that Q(x)

• When true

• When false

Translation Examples

x, x20x, x2-1 m such that m2=m

Translation Examples

• If a real number is an integer then it is a rational number

• All bytes have eight bits

• No fire trucks are green.

Formal and Informal

• For all polygons p, if p is a square, then p is a rectangle.

• For all squares p, p is a rectangle.

• There exists a number n such that n is prime and n is even.

• There exists a prime number n such that n is even.

Trailing Quantifier

• For all squares p, p is a rectangle.

• p is a rectangle for any square p.

• There exists a prime number n such that n is even.

• n is even for some prime number n.

Equivalent forms of and

xU if P(x) then Q(x) xD, Q(x)

Where D is all x such that P(x)

Implicit Quantification

• If n is a number, then it is a rational number

• If x>2 then x2>4.

• x>2 x2>4

and

• P(x) Q(x) means that the truth set of P(x) is contained in the truth set of Q(x).

• P(x) Q(x) means P(x) and Q(x) have identical truth sets.

Negation of Quantified Statements

• For all x in D, Q(x)

• There exists x in D such that ~Q(x).

• “all are” versus “some are not”

Negation of Quantified Statements

• There exists x in D such that Q(x)

• For all x in D, ~Q(x).

• “some are” versus “all are not”

Try These

• ~(For every prime p, p is odd)

• ~(There exists a triangle T, such that the sum of the angles of T equals 200 degrees)

No Politicians are Honest

• Formal

• Formal negation

• Informal negation

All computer programs are finite.

• Formal

• Formal negation

• Informal negation

Some dancers are over 40

• Formal

• Formal negation

• Informal negation

~(x, P(x)Q(x))

x such that ~(P(x)Q(x))x such that P(x)~Q(x)

More on Universal Conditional

xU if P(x) then Q(x)

• Contrapositive

• Converse

• Conditional

Necessary and Sufficient Conditions, Only If

x, r(x) is a sufficient condition for s(x). x, r(x) is a necessary condition for s(x). x, r(x) only if s(x).