1. UNIVERSITI PENDIDKAN SULTAN IDRIS
SUBTOPIC 3
QUANTIFIERS
PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT

3. Introduction
A proposition is a statement; either “true” or “false”.
The statement
P : “n” is odd integer.
The statement P is not proposition because whether p is
true or false depends on the value of n

5. Topic
1 • Quantifiers
2 • Universal Quantification
• Counterexample
3
• Existential Quantification
4
5 • De Morgan’s Law For Logic

7. 1. Quantifiers
Definition:
Let P (x) be a statement involving the variable
x and let D be a set.
We called P a proportional function or predicate
(with respect to D ) , if for each x ∈ D , P (x) is a
proposition.
We called D the domain of discourse of P.

9. Example 1
Let P(n) be the statement
n is an odd integer
For example:
If n = 1, we obtain the proposition.
P (1): 1 is an odd integer (Which is true)
If n = 2, we obtain the proposition
P (2): 2 is an odd integer (Which is false)

11. 2. Universal Quantification
Definition:
Let P be a propositional function with the domain of
discourse D. The universal quantification of P (x) is the
statement. “For all values of x, P is true.”
∀x, P (x)
Similar expressions:
For each…
For every…
For any…

13. 3. Counterexample
Definition :
A counterexample is an example chosen to show that a
universal statement is FALSE.
To verify :
∀x, P (x) is true
∀x, P (x) is false

15. Example 2

17. 4. Existential Quantification
Let P be a proportional function with the domain of
discourse D. The existential quantification of P (x) is the
statement. “there exist a value of x for which P (x) is
true.
∃x, P(x)
Similar expressions :
- There is some…
- There exist…

19. Example 3

21. 5. De Morgan’s Law For Logic
Theorem:
(∀x, P (x)) ≡ (∃x, (P(x))
(∃x, (P(x)) ≡ (∀x, P (x))
The statement
“The sum of any two positive real numbers is
positive”.
∀x > 0∀y > 0

23. Example 4