‫‪MATH 226‬‬
‫رياضيات متقطعة لعلوم الحاسب‬
Chapter 1
Predicates and Quantifiers
1.4
Predicates
Statements involving variables, such as
“x > 3,” “x = y + 3,” “x + y = z,”
and
“computer x is functioning properly,”
Variables
A variable is a symbol that stands for an individual in a
collection.
For example, the variable x may stand for one of the days. We
may let x = Monday or x = Tuesday, etc.
A collection of objects is called the domain of a variable.
For the above example, the days in the week is the domain of
variable x.
Incomplete Statements
A sentence containing a variable is called an incomplete
statement.
When we replace the variable by the name of an individual in
the set we obtain a complete statement.
Example : incomplete statement : “x has 30 days.”
Here, x can be any month and substituting that, we will get a
complete statement.
“ April has 30 days.”
 The statement “x is greater than 3” has two parts.
 The first part, the variable x,
 The second part the predicate, “is greater than 3”—
refers to a property that
 We can denote the statement “x is greater than 3” by P(x),
Truth value for predicate
EXAMPLE : Let P(x) denote the statement “x > 3.” What are
the truth values of P(4) and P(2)?
Solution:
P(4), is the statement “4 > 3,” is true.
However, P(2), is the statement “2 > 3,” is false.
 Quantifiers:
 Quantifiers : refer to given quantities,
 The area of logic that deals with predicates
and quantifiers is called the predicate calculus.
Two kinds of quantifiers: Universal and Existential
 Universal Quantifier:
represented by ) ∀) The symbol is translated as and means
“for all”, “given any”, “for each,” or “for every,” and is known
as the universal quantifier.
 Existential Quantifier:
represented by ) ∃( The symbol is translated as “for some,”
“there exists,” “there is a,” or “for at least one”.
“There is an x such that P(x),” or “There is at least one x
such that P(x),” or“For some x P(x).”
Truth of expression ∀x P(x)
 Example :“P(x) be the statement “x + 1 > x.” What is the
truth value of the quantification ∀xP(x),where the domain
consists of all real numbers?
Because P(x) is true for all real numbers x, the
quantification ∀xP(x) is true.
 Example :P(x) is the property that x is a plant, and the
domain is the collection of all flowers: is true.
 Example P(x) is “x*2 > 0.” show that ∀xP(x) is false where
the universe of discourse consists of all integers
 we give a counterexample. We see that x = 0 is a
x*2 = 0 when x = 0, so that x2 is not greater than 0 cause
when x = 0.
Truth of expression ∃xP(x)
 EXAMPLE 5 : Let P(x) denote “x > 3.” What is the truth
value of the quantification ∃xP(x),where the domain
consists of all real numbers?
 Solution: Because “x > 3” is sometimes true—for
instance, when x = 4 ∃xP(x), is true.
 That is, ∃xP(x) is false if and only if P(x) is false for
every element of the domain.
 EXAMPLE 6: Let Q(x) :“x = x + 1.”What is the truth
value ∃xQ(x), the domain consists of all real numbers?
 Solution: Because Q(x) is false for every real number
,∃xQ(x) is false.
Negating Quantified Expressions
• consider the negation of the statement
“Every student in your class has taken a course in calculus.”
the domain is the students in your class.

• The negation of this statement is “It is not the case that
every student in your class has taken a course in
calculus.”
This is equivalent to “There is a student in your class who has
not taken a course in calculus.”
And this is simply, ∃x ￢P(x).
• This example illustrates that :￢∀xP(x) ≡ ∃x ￢P(x).
 consider the proposition“
 There is a student in this class who has taken a course in
calculus.” This is the existential quantification ∃xQ(x),
 The negation of this statement is “It is not the case that
there is a student in this class who has taken a course in
calculus.” This is equivalent to “Every student in this
class has not taken calculus,” which is just ∀x ￢Q(x).
 This example illustrates the equivalence
￢∃xQ(x) ≡ ∀x ￢Q(x).
Translating from English into Logical Expressions
 Example : Express the statement “Every student in this class has
studied calculus” using predicates and quantifiers.
 Solution:



introduce a variable x “For every student x in this class, x has
studied calculus.”
If S(x) represents the statement that (person x is student in
this class) we introduce C(x): “x has studied calculus )and
domain all people, we will need to express our statementas
“For every person x, if person x is a student in this class then x
has studied calculus.”” ∀x(S(x) → C(x)).
Not as ∀x(S(x) ∧C(x)) because this statement says that all
people are students in this class and have studied
 Example : Express the statements “Some student in this
class has visited Mexico” using predicates and quantifiers.
Solution:
the domain for the variable x consists of all people ,
 We introduce M(x), which is the statement “x has visited
Mexico.”,.
 We introduce S(x) to represent “x is a student in this
class.” Our solution is ∃x(S(x) ∧ M(x)) because the
statement is that there is a person x who is a student in
this class and who has visited Mexico.
 statement cannot be expressed as ∃x(S(x) → M(x))
why ?
Example :
•
“Every person is nice” domain : any thing, if it is a person, then it is
nice.”
• So, if P(x) is “x is a person” and Q(x) be “x is nice,” the statement can be
symbolized as
(∀ x)[P(x) → Q(x)]
• All persons are nice” or “Each person is nice” will also have the same
symbolic form.
Example
 “There is a nice person” can be rewritten as “There exists something
that is both a person and nice.”
 In symbolic form, (∃ x([P(x( Λ Q(x(].
“Some persons are nice” or “There are nice persons.” will also have the
same symbolic form.
 so almost always, ∃ goes with Λ (conjunction) and ∀ goes with →
(implication