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8
Predicates and Quantifers
8.1
Text references
•
we are omitting
§
1.4 and
§
1.5 in Epp
•
some material in there that may prove useful in other courses
gates and circuits — computer organization courses
representation of numbers in diﬀerent bases — possibly APS105
she looks in detail at the representation of integers of size
byte
•
we are moving on to Chapter 2
in that chapter, we will be omitting “Tarski’s World”
8.2
Predicates
•
recall that a statement is a sentence that is either true or false (but not both)
•
recall that a sentence such as “
x
is less than 10” is not a statement
why? — because truth value depends on
x
•
we call such a sentence a
predicate
area of logic dealing with predicates is called
predicate calculus
•
we can write predicates symbolically
examples
we could write “
x
is less than 10” as
L
(
x
)
we could write “
x
is greater than
y
”as
G
(
x,y
)
we could write “
x
is a student at U of T” as
S
(
x
)
•
we can convert a predicate to a statement by assigning a value to the variable(s)
examples using preceding predicates
e.g.
L
(12) is a false statement
8.3
Truth sets of predicates
•
in dealing with variables, there is always some domain of discourse
e.g.
Z
,
R
, ECE students, human beings, etc.
•
the
truth set
of a predicate is the set of elements in the domain for which the predicate is true
•
as an example, if
P
(
x
) is the predicate “
x
2
is less than or equal to 25” and the domain is
Z
+
then we could write the truth set of
P
(
x
)as
{
x
∈
Z
+

P
(
x
)
}
here, the value of the truth set is
{
1
,
2
,
3
,
4
,
5
}
8.4
The universal quantiFer,
∀
•
another way to obtain a statement from a predicate is to use a
quantifer
•
suppose domain of discourse is
S
=
{
1
,
2
,
3
}
consider the predicate
P
(
x
): “
x
2
<
10”
•
if we substitute any value from the domain, we always get a true statement
i.e.
P
(1)
∧
P
(2)
∧
P
(3) is true
•
we can write this symbolically as:
∀
xP
(
x
)
note that this is a
statement
, not a
predicate
•
more generally, given a domain
D
, if a predicate is true for every value in
D
, we write
∀
x
∈
D, P
(
x
)
or
∀
x
∈
D
(
P
(
x
))
17
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we read this as
“For all
x
in
D
,
P
(
x
) is true”
•
sometimes written using conditional
∀
x
,if
x
∈
D
then
P
(
x
)
or
∀
x
(
x
∈
D
→
P
(
x
))
this is known as a
universal conditional
statement
•
if domain of discourse is understood, we may simply write
∀
x
(
P
(
x
))
8.5
The existential quantifer,
∃
•
sometimes we want to assert that a predicate is true (at least) some of the time
•
e.g.
at least one person in the room owns a dog
•
let domain of discourse be
R
, people in this room
let
D
(
x
)be“
x
owns a dog”
•
we can write the statement symbolically in the form
∃
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This note was uploaded on 04/19/2008 for the course ECE 190 taught by Professor Carter during the Fall '06 term at University of Toronto Toronto.
 Fall '06
 Carter
 Gate

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