CSE

1. Discrete Mathematics
Lecture 07
Rubya Afrin
Lecturer
Northern University of Business and Technology

2. Universal Quantification
Let P(x) be a propositional function.
Universally quantified sentence:
For all x in the universe of discourse P(x) is true.
Using the universal quantifier :
x P(x) “for all x P(x)” or “for every x P(x)”
(Note: x P(x) is either true or false, so it is a proposition, not a
propositional function.)
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3. Universal Quantification
Example:
S(x): x is a NUB student.
G(x): x is smart.
What does x (S(x)  G(x)) mean ?
“If x is a NUB student, then x is a smart.”
or
“All NUB students are smart.”
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4. Existential Quantification
Existentially quantified sentence:
There exists an x in the universe of discourse for which P(x) is true.
Using the existential quantifier :
x P(x) “There is an x such that P(x).”
 “There is at least one x such that P(x).”
(Note: x P(x) is either true or false, so it is a proposition, but no
propositional function.)
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5. Existential Quantification
Example:
P(x): x is a NUB professor.
G(x): x is a genius.
What does x (P(x)  G(x)) mean ?
“There is an x such that x is a NUB professor and x is a genius.”
or
“At least one NUB professor is a genius.”
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6. Quantification
Another example:
Let the universe of discourse be the real numbers.
What does xy (x + y = 320) mean ?
“For every x there exists a y so that x + y = 320.”
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Is it true?
Is it true for the natural numbers?
yes
no

7. Disproof by Counterexample
A counterexample to x P(x) is an object c so that P(c) is false.
Statements such as x (P(x)  Q(x)) can be disproved by simply
providing a counterexample.
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Statement: “All birds can fly.”
Disproved by counterexample: Penguin.

8. Negation
 (x P(x)) is logically equivalent to x (P(x)).
 (x P(x)) is logically equivalent to x (P(x)).
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9. Negation

10. Examples
 Let 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?
 Solution: Because P (x) is true for all real numbers x, the quantification ∀xP (x) is
true.
 Let P(x) denote the statement “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—the
existential quantification of P(x), which is ∃xP(x), is true.

11. Negating Quantified Expressions
 “Every student in your class has taken a course in Discrete Math.”
 This statement is a universal quantification, namely, ∀xP (x),
where P (x) is the statement “x has taken a course in calculus” and the domain
consists of the students in your class.
 The negation of this statement is
 ∃x ¬P (x).
 “There is a student in your class who has not taken a course in Discrete Math.”
 This example illustrates the following logical equivalence:
¬∀xP (x) ≡ ∃x ¬P (x).

12. Question???

13. Thank you…