1.3 predicates and quantifiers. chapter 1, section 3 predicates and quantifiers predicate e.g., if...
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1.3 Predicates and Quantifiers
Chapter 1, section 3Predicates and Quantifiers
Predicate
• E.g., “If it is sunny, I’ll buy X.” Here the parameter is X.
Chapter 1, section 3Predicates and Quantifiers
Quantifiers
• Universal quantification
• Existential quantification
Chapter 1, section 3Predicates and Quantifiers
Universal Quantification
• Notation: ( x P(x))– This is read: “for all x P(x) is true.”
• This could be used to express the concept:– “Every sunny day I buy a red bag.”
Chapter 1, section 3Predicates and Quantifiers
Example, universal quantification
• C(x) == "x has taken algebra”D(x) == "x is enrolled in discrete math”
x (D(x) C(x)) {is True} but
x (C(x) D(x)) {is False}
Chapter 1, section 3Predicates and Quantifiers
Existential Quantification
• Notation: ( x P(x))– This is read “there exists an x such that
P(x) is true”
• This could be used to express the concept that at least once I bought a red bag.
Chapter 1, section 3Predicates and Quantifiers
Example, existential quantification
x (C(x) D(x))
• This is TRUE – if we can find one person who has taken
algebra – AND is enrolled in discrete math, – OR find one person who hasn't taken algebra.
Chapter 1, section 3Predicates and Quantifiers
From English to logical expressions
• Equivalent to :– there exists
– there is
– there is at least one
– there is some
– for some
– some
– for at least one
• Equivalent to :– for all
– all
– any for every
– every for any any
– for arbitrary
– an arbitrary
– for each
Chapter 1, section 3Predicates and Quantifiers
Binding Variables
• E.g., y x P (x,y,z)
– bound variables y and x,
– and free variable z.
Chapter 1, section 3Predicates and Quantifiers
Consider:
z y x P(x,y,z)
Chapter 1, section 3Predicates and Quantifiers
More Examples
• Suppose P(x,y,z) is the predicate
“When I teach discrete math in semester x, student y does well on exam z.”
• Then x y z P(x,y,z) is the statement:
• Every time I teach discrete math, there is at least one student who does well on every exam."
Chapter 1, section 3Predicates and Quantifiers
Order matters!
y x z P(x,y,z)
Chapter 1, section 3Predicates and Quantifiers
Negation of quantifiers:
~ x P(x) x ~P(x)– true:
• P(x) is false for every x.
– false: • There is an x for which
P(x) is true.
~ x P(x) x ~ P(x)– true:
• There is an x for which P(x) is false.
– false: • P(x) is true for every x.
Chapter 1, section 3Predicates and Quantifiers
Self Quiz
• Simplify the following by moving “~” inside the quantifiers and connectors:
~ x y z ( P(x) V ( Q(y) R(z)))
Chapter 1, section 3Predicates and Quantifiers
Answer:
~ x y z ( P(x) V ( Q(y) R(z)))
x y z (~ P(x) (~ Q(y) V ~ R(z)))