This document defines and provides examples of quantifiers - universal and existential quantification. Universal quantification uses "for all" and is represented by ∀, while existential quantification uses "there exists" and is represented by ∃. A counterexample can disprove a universal statement by showing a case that makes the proposition false. De Morgan's laws state that the negation of a universal statement is an existential statement, and vice versa. Examples are provided to illustrate these concepts.

1. SUBTOPIC 3 : QUANTIFIERS
The statement
P: n is odd integer.
A proposition is a statement that is either true or false. The statement p is not proposition
because whether p is true or false depends on the value of n. For example, p is true if n = 104 and
false if n = 8. Since, most of the statements in mathematics and computer a science use variable,
we must extend the system of logic to include such statements.
1.Quantifiers
Definition:
Let P (x) be a statement involving the variable x and let D be a set. We call P a proportional
function or predicate (with respect to D ) , if for each x ∈ D , P (x) is a proposition. We call
D the domain of discourse of P.
Example 1: Let P(n) be the statement
n is an odd integer
Then P is a propositional function with the domain of discourse since for each n ∈
, P(n) is a proposition [for each n ∈ , P(n) is true or false but not both]. 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)
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2. A propositional function P, by itself, is neither true nor false. However, for each x is
domain of discourse, P (x) is a proposition and is, therefore, either true or false. We can think of
propositional function as defining a class of propositions, one for each element in the domain of
discourse. For example, if P is a propositional function with domain of discourse , we obtain
the class of propositions.
P (1), P (2), …..
Each of P (1), P (2), …. Is either true or false.
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…
3. Counterexample
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
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3. Example 1:
Consider the universally quantified statement.
∀x ( ≥ 0)
The domain of discourse is R. The statement is true because for every real number x, it is true
that the square of x is positive or zero.
According the definition, the universally quantified statement.
∀x, P (x)
Is false for at least one x in the domain of discourse that makes P (x) is false. A value x in the
domain of discourse that makes P (x) false is called a counterexample to the statement.
Example 2:
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
Is false. The value1 is counterexample of the statement.
∀x ( -1 ≥ 0)
Although there are values of x that make the propositional function true, the counterexample
provide show that the universally quantified statement is false.
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4. 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 exists a value of x for which P (x) is true.
∃x, P(x)
Similar expressions:
- There is some…
- There exist…
There is at least…
Example 1:
Consider the existentially quantified statement.
∃x (
The domain of discourse is R. the statement is true because it is possible to find at least one real
number x for which the proposition
Is true. For example, if x = 2, we obtain the true proposition.
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5. Is not the case that every value of x results in a true proposition. For example, if x = 1, the
proposition
Is false.
According to definition, the existentially quantified statement
∃x, P(x)
Is false for every x in the domain of discourse, the proposition P (x) is false.
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6. 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,
Example 1: Let P(x) be the statement
We show that
∃x, P(x)
Is false by verifying that
∀x, ⌐ P (x)
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.
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7. 1. EXERCISE.
a) let P (x) be the propositional function “x ≥ .” Tell whether each proposition is
true or false. The domain of discourse is R
i. P (1)
ii. ⌐∃x P(x)
b) Suppose that the domain of discourse of the propositional function P is {1, 2, 3,
4}. Rewrite each propositional function using only negation, disjunction and
conjunction.
i. ∀x P (x)
c) Determine the truth value, the domain discourse R x R, justify the answer.
i. ∀x ∀y ( < y + 1)
d) Assume that ∀x ∃y P(x, y) is false and that the domain of discourse is nonempty.
It must also be false? Prove the answer.
i. ∀x ∀y P (x,y)
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