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SUBTOPIC 3QUANTIFIERS
UNIVERSITI PENDIDKAN SULTAN IDRIS
PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT
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
Introduction
Topic
1 • Quantifiers
2• Universal
Quantification
3 • Counterexample
4• Existential
Quantification
5• De Morgan’s Law
For Logic
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.
1. Quantifiers
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)
Example 1
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…
2. Universal Quantification
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
3. Counterexample
Consider the universally quantified statement.∀x (-1 ≥ 0)
The domain of discourse is R. The statement is false since, if x = 1, the proposition
-1 >0
It is false. The value 1 is counterexample of the statement.∀x (-1 ≥ 0)
Although there are values of x that make the propositional function true, the counterexample shows that the universally quantified statement is false.
Example 2
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…
4. Existential Quantification
Consider the existentially quantified statement.∃x (
True. For example, if x = 2, we obtain the true proposition.
It is true because it is possible to find at least one real numberx for which the proposition
False. For example, if x = 1, the proposition
It is false for every x in the domain of discourse, the proposition P(x) is false
Example 3
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
5. De Morgan’s Law For Logic
Let P(x) be the statement
We show that ∃x, P(x) It is false by verifying that∀x, ⌐ P (x) It is true. The technique can be justified by
appealing to theorem. After we prove that proposition is true, we may negate and conclude that is false.
By theorem, ∃x, ⌐⌐P(x) or equivalently ∃x, P(x) is also false.
Example 4