Formal Logic - Lesson 8 - Predicates and Quantifiers

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FORMAL LOGIC
Discrete Structures I
FOR-IAN V. SANDOVAL

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Lesson 8
PREDICATES AND QUANTIFIERS

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LEARNING OBJECTIVES
❑ Use the formal symbols of predicate logic
❑ Determine the truth value in some interpretation of an
expression in predicate logic
❑ Use predicate logic to represent English language
sentences

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PREDICATE
❑ A predicate (or open statements) is a statement whose truth
depends on the value of one or more variables.
❑ Predicate become propositions once very variable is bound
by assigning a universe of discourse.
❑ Most of the propositions are define in terms of predicates.
❑ Predicates: statements involving variables, e.g., “x > 3”,
“x=y+3”, “x+y=z”, “computer x is under attack by an
intruder”, “computer x is functioning property”

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PREDICATE
❑ Example: x > 3
- The variable x is the subject of the statement
- Predicate “is greater than 3” refers to a property that the
subject of the statement can have
- Can denote the statement by P(x) where P denotes the
predicate “is greater than 3” and x is the variable
- P(x): also called the value of the propositional function
P at x
- Once a value is assigned to the variable x, P(x) becomes
a proposition and has a truth value

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PREDICATE
❑ Example
“If x is an odd number, then x is not multiple of 2.”
- Logical form P(x) → Q(x)
❑ Example
“There exists an x such that x is odd number and 2x is
even number.”
“For all x, if x is a positive integer, then 2x + 1 is and odd
number”
- Cannot be represented using logical connectives

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Take Note:
❑ Binding variable
- used on the variable x; the occurrence of the variable is
bound
- a variable is said to be free; if the occurrence of a
variable is not free
❑ To convert a propositional function into a proposition
- all variable must be bound or a particular value must be
designated to it
- it is done by applying a combination of quantifiers
(universal, existential) and the value assignments
❑ Scope of a quantifier
- the part of an assertion in which variables are bound by
the quantifiers.

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QUANTIFIERS
❑ Express the extent to which a predicate is true
❑ In English, all, some, many, none, few
❑ Focus on two types:
- Existential Quantifiers
- Universal Quantifiers
❑ Predicate Calculus
- the area of logic that deals with predicates and quantifiers

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Existential Quantifiers
❑ a predicate is true for there is one or more elements under
consideration
❑ “There exists an element x in the domain such that P(x) (is
true)”
❑ Denote that as ∃x P(x) where ∃ is the existential quantifier
❑ In English, “for some”, “for at least one”, or “there is”
❑ Read as “There is an x such that P(x)”, “There is at least
one x such that P(x)”, or “For some x, P(x)”

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❑ Example
Let P(x) be the statement “x > 3”. Is ∃x P(x) true for the
domain of all real numbers?;
Let Q(x) be the statement “x=x+1”. Is ∃x P(x) true for the
domain of all real numbers?
- When all elements of the domain can be listed, , e.g.,
x1, x2, …, xn,
- it follows that the existential quantification is the same
as disjunction P(x1) v P(x2) … v P(xn)

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❑ Example
What is the truth value of ∃x P(x) where P(x) is the
statement “x2 >10” and the domain consists of positive
integers not exceeding 4?
- ∃x P(x) is the same as P(1) v P(2) v P(3) v P(4)

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Universal Quantifiers
❑ “P(x) for all values of x in the domain”
❑ Denote that as ∀x P(x) where ∀ is the universal quantifier
❑ The statement “∀x P(x)” is true of every value of x
❑ Read it as “for all x P(x)” or “for every x P(x)”
❑ A statement ∀x P(x) is false if and only if P(x) is not always
true
❑ An element for which p(x) is false is called a
counterexample of ∀x P(x)

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❑ Example
Let p(x) be “x2>0”. To show that the statement ∀x P(x) is
false where the domain consists of all integers. Show a
counterexample with x=0
- When all the elements can be listed, e.g., x1, x2, …, xn
- it follows that the universal quantification ∀x P(x) is the
same as the conjunction P(x1) ˄ P(x2) … ˄ P(xn)

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❑ Example
What is the truth value of ∀x P(x) where P(x) is the
statement “x2 < 10” and the domain consists of positive
integers not exceeding 4?
- ∀x P(x) is the same as P(1) ˄ P(2) ˄ P(3) ˄ P(4)

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Unique Existential Quantifiers
❑ there exists a unique x such that P(x) is true ∃! P(x)
❑ denote that as ∃! P(x) where ∃! is called the Unique
❑ The statement ∃! P(x) is true for one and only x in the
universe discourse.
❑ The statement ∃! P(x) is true for one and only x in the
universe discourse.

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❑ Common Words Associated with various Connectives

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Quantifiers with Restricted Domains
❑ What do the following statements mean for the domain of
real numbers?
∀x < 0, x2 > 0 same as ∀x (x < 0 → x2 > 0)
∀y ≠ 0, y3 ≠ 0 same as ∀y (y ≠ 0 → y3 ≠ 0)
∃z > 0, z2 = 2 same as ∃z (z > 0 ˄ z2 = 2)
❑ Be careful about → and in these statements

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Precedence of Quantifiers
❑ ∀ and ∃ have higher precedence than all logical operators
from propositional calculus
∀x P(x) v Q(x) ≅ (∀x P(x)) v q(x) rather than
∀x (P(x) v Q(x))

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Truth Values of Quantifiers
❑ If the universe of discourse for P is P(p1, p2, ..., pn), then ∀x
P(x)↔P(p1)^P(p2)^…^P(pn) and ∃x P(x)↔ P(p1) P(p2) v
P(Pn).

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Negating Quantifiers
❑ Consider:
“All students in this class have red hair.”
❑ What is required to show the statement is false?
- There exists a student in this class that does NOT have
red hair.
❑ To negate a universal quantification:
- negate the propositional function
- AND change to an existential quantification
~∀x P(x) ≅ ∃x ~P(x)

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Negating Quantifiers
❑ Consider:
“There is a student in this class with red hair.”
❑ What is required to show the statement is false?
- All students in this class do not have red hair
❑ To negate an existential quantification:
- negate the propositional function
- AND you change to a universal quantification
~∃x P(x) ≅ ∀x ~P(x)

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Quantification and Conjunction
❑ Predicates can be joined by the logical operations

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Example 1:
Write the following statements as a logical expression.
a. Every child has biological parents.
Solution:
Let P(x , y)= “x is biological parent of y.”
Logical Expression:
∀y ∃x ∀z P(x, y) (P(x, y) [(z ≠ x) → ~ P(z, y)]

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Examples:
b. There is a number x such that when if is added to any
number , the result is that number , and if it is multiplied in
any number, the result is x.
Solution:
Let P(x , y) = “x +y = y”
Let Q(x, y) = “xy = x”
Logical Expression:
∃x ∀y (P(x , y) ^ (P(x, y))

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c. Not everybody can drive a car.
Solution:
Let P(x) = “x can drive a car.”
The statement “everybody can drive a car”, can be
expressed as ∀x P(x).
The negation ~ ∀x P(x), we can also say that “There is
somebody that cannot drive a car.” which can be expressed as
∃x ~P(x).

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d. Nobody can swim.
Solution:
Let P(x) = “x can swim.”
The statement “somebody can swim”, can be expressed as
∃x P(x).
The negation ~∃x P(x), another way to state this is
“Everybody cannot swim”, which can be expressed as ∀x
~P(x).

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Example 2:
Let P(x) = “x + y=9” where the universe of discourse for x
and y is the subset of integers. Write the logical expression of
the following statements and denote its truth values.
a. For every x and every y, x + y > 9
b. For every x, there is some value y such that x + y > 9
c. There is a value of y such that for every x, x + y >9

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a. For every x and every y, x + y > 9
Solution:
The statement can be expressed as ∀x ∀y P(x , y).
If we let x = 1 and y = 1, P(1,1) is clearly false, it follows that
∀x ∀y P(x , y) is false.

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b. For every x, there is some value y such that x + y > 9
Solution:
Statement can be expressed as ∀x ∃y P(x, y)
For any x, if we choose y = x + (-x+10) > 9 = “10>9”, which
is clearly true, so ∀x ∃y P(x, y) is also true.

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c. There is a value of y such that for every x, x + y >9
Solution:
Statement can be expressed as ∃x ∀y P(x, y)
If x = -y, P(x , y) = “x+(-x)>9” = “0>9” which is false.
This means that we cannot find any y which works with all
x, so ∃x ∀y P(x, y) is also false.

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Predicate Logic to Represent English Language Sentences
❑ Example 1
Consider “For every student in this class, that student has
studied Calculus.”
• Rephrased: For every student x in this class, x has
studied Calculus.
- Let C(x) be “x has studied Calculus”
- Let S(x) be “x is a student”
• ∀x C(x)
- True if the universe of discourse is all students in this
class

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❑ Example 1
Consider “For every student in this class, that student has
studied Calculus.”
• What about if the universe of discourse is all students (or
all people?)
- ∀x (S(x) → C(x))
• Another Option
- Let Q (x, y) be “x has studied y”
- ∀x (S(x) → C(x, calculus))

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❑ Example 2
“Some students have visited Mexico.”
• Rephrased: “There exists a student who has visited
Mexico.”
- Let S(x) be “x is a student”
- Let M(x) be “x has visited Mexico”
• ∃x M(x)
- True if the universe of discourse is all students
• ∃x (S(x) M(x))
- If the universe of discourse is all people

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❑ Example 3
“Every student in this class has visited Canada or Mexico.”
• Let
- S(x) be “x is a student”
- M(x) be “x has visited Mexico”
- C(x) be “x has visited Canada”
• ∀x (M(x)∨C(x))
- When the universe of discourse is all students
• ∀x (S(x)→(M(x)∨C(x))
- When the universe of discourse is all people

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❑ Example 3
“Every student in this class has visited Canada or Mexico.”
• Other option
- S(x) be “x is a student”
- Let V(x, y) as “x has visited y”
• ∀x (S(x)→(V(x,Mexico) ∨ V(x,Canada))
- When the universe of discourse is all people

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GROUP ENRICHMENT ACTIVITY
1. Determine the universe of discourse in the statement “x
has fur”.
2. Express the statements “Everybody in this class is a
student in this school”, as logical expression.
3. If P(x) = “x < 0” what are the truth values of P(0) and P(1).
4. Given P(x) = “x must enroll in Discrete Structure”
Q(x) = “x is an IT student”
Write the logical expression of
a. Every IT students must take Discrete Structure
b. Everybody must take Discrete Structure
c. Not everybody must tale Discrete Structure

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GROUP ENRICHMENT ACTIVITY
5. Consider
– “All hummingbirds are richly colored”
– “No large birds live on honey”
– “Birds that do not live on honey are dull in color”
– “Hummingbirds are small
Translate the statements:
a. “All hummingbirds are richly colored”
b. “No large birds live on honey”
c. “Birds that do not live on honey are dull in color”
d. “Hummingbirds are small”

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• Levin, O. (2019). Discrete Mathematics: An Open Introduction 3rd Edition. Colorado: School of Mathematics Science
University of Colorado.
• Aslam, A. (2016). Proposition in Discrete Mathematics retrieved from https://www.slideshare.net/AdilAslam4/chapter-1-
propositions-in-discrete-mathematics
• Blaircomp (2014). Predicates and Quantifiers. Retrieved from https://www.slideshare.net/blaircomp2003/predicates-
and-quantifiers
• Operator Precedence retrieved from http://intrologic.stanford.edu/glossary/operator_precedence.html
REFERENCES

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