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