Page 10 - Chapter 2

• Existential Quantifiers (∃)

The Existential Quantifiers of a predicate P(x) is the statement

“there exists a value of x for which P(x) is true”, denoted by

Example 9 ∃ x P(x)

(a) Let Q(x): x+1<4. then the existential quantification of

Q(x), ∃ x Q(x), is a true statement, since Q(1) is a true

statement

(b) The statement ∃ y y+2=y is false since there is no value of y

for which the propositional function y+2=y produces a true

statement.
• The order of the Quantifiers ∀ & ∃

The order does not affect the output for the same

quantifiers, while it may produce different results for different

quantifiers.

E.g.

P(x, y) : x + y =1 P(x, y): x *y = 0

∀ x ∃ y P(x) is true, ∃ y ∀ x is false. ∀ x ∃ y P(x) is true, ∃ y ∀ x is true too

.