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).