Introduction
Predicates and Quantifiers II
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Predicates and Quantifiers II
82
Previous Lecture
Predicates
Assigning values, universe, truth values
Quantifiers
Discrete Mathematics – Predicates and Quantifiers II
83
Quantifiers and Negations
Summarizing
true
false
2200
x P(x)
5
x P(x)
Observe that
2200
x P(x)
is false if and only if
5
x
¬
P(x)
is true
5
x P(x)
is false if and only if
2200
x
¬
P(x)
is true
For every value
a
from
the universe
P(a) is true
There is a counterexample –
a value
a
from the universe
such that
P(a)
is false
There is a value
a
from
the universe such that
P(a)
is true
For all values
a
from the
universe
P(a)
is false
Discrete Mathematics – Predicates and Quantifiers II
84
Example
What is the negation of each of the following statements?
Statement
Negation
All lions are fierce
2200
x P(x)
There is a peaceful lion
Everyone has two legs
2200
x P(x)
There is a person having more
than two legs, one leg, or no legs
at all
Some people like
coffee
5
x P(x)
All people hate coffee
There is a lady in
one of these rooms
5
x P(x)
There is a tiger in every room
(Some rooms contains
a lady)
Discrete Mathematics – Predicates and Quantifiers II
85
Multiple quantifiers
Often predicates have more than one variable. In this case we
need more than one quantifier.
P(x,y)
=
``car
x
has colour
y’’
2200
x
2200
y
P(x,y)
``every car is painted all colours
5
x
5
y
P(x,y)
``there is a car that is painted some colour
2200
x
5
y
P(x,y)
``every car is painted some colour
5
x
2200
y
P(x,y)
``there is a car that is painted all colours
Discrete Mathematics – Predicates and Quantifiers II
86
Open and Bound Variables
In the statement
``car
x
has some colour’’
5
y P(x,y)
variables
x
and
y
play completely different roles.
Variable
y
is
bound
by the existential quantifier.
Effectively it disappeared from the statement.
Variable
x
is not bound, it is
free
.
Another example:
``x is the least number’’
Q(x,y) = ``x
is less than
y’’
2200
y
Q(x,y)
or
2200
y (x
≤
y)
``x is the greatest number’’
2200
y
Q(y,x)
or
2200
y (y
≤
x)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Discrete Mathematics – Predicates and Quantifiers II
87
Defining New Predicates
Quantifying some variables helps creating new predicates
``car
x
has some colour’’
Q(x) =
5
y P(x,y)
Let
P(x,y) mean
``lion
x
likes
y’’
Q(y) =
2200
x
P(x,y)
``every lion likes
y’’
Q(meat)
is true
Q(apples)
is false
R(x) =
2200
y
P(x,y)
``lion
x
likes everything’’
R(x)
is always false
(say it using quantifiers:
2200
x
¬
R(x)
or
2200
x
¬
(
2200
y
P(x,y))
)
S(x) =
5
y
P(x,y)
``lion
x
has favorite food’’
Discrete Mathematics – Predicates and Quantifiers II
88
Quantifiers and Compound Statements
Quantifiers can be used together with logic connectives
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 AndreiBulatov
 Logic, Quantification, Universal quantification, Quantifiers II

Click to edit the document details