Understanding the Existential Quantification (∃x) in Mathematical Logic | Explained with Examples

∃x P(x)

The mathematical expression ∃x P(x) represents the existential quantification of a predicate P(x)

The mathematical expression ∃x P(x) represents the existential quantification of a predicate P(x). This notation is used in first-order logic to denote that there exists at least one element x in the domain of discourse that satisfies the predicate P(x).

In simpler terms, ∃x P(x) asserts the existence of an element x for which the property P holds true. The domain of discourse refers to the set of elements under consideration in a particular context.

For example, if the domain of discourse is the set of all integers, and P(x) represents the property “x is even,” then ∃x P(x) would be read as “There exists an integer x that is even.” This statement is true because the set of integers contains even numbers.

On the other hand, if P(x) represents the property “x is prime,” then ∃x P(x) would be read as “There exists an integer x that is prime.” In this case, the truth of the statement depends on the domain of discourse. If the domain includes the set of prime numbers, then the statement is true. However, if the domain is limited to the set of even numbers, then the statement is false.

In conclusion, the expression ∃x P(x) is used in mathematical logic to assert the existence of an element x that satisfies a given property P.

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