Logical Implication | If There Exists an Element x such that P(x) is True, Then P(c) is True for Some Element c

∃xP(x)___________∴P(c) for some element c

The given statement is a logical implication, which can be read as “If there exists an element x such that P(x) is true, then P(c) is true for some element c

The given statement is a logical implication, which can be read as “If there exists an element x such that P(x) is true, then P(c) is true for some element c.”

To understand this statement, let’s break it down:

∃xP(x): This is an existential quantifier (∃) followed by a statement P(x). It means that there exists at least one element x for which the statement P(x) is true. In other words, there is at least one element in the domain that satisfies the property P.

∴: This symbol means “therefore” or “implies.” It represents a logical implication where the statements before it logically lead to the statement after it.

P(c): This is a replacement of variable x with a specific element c. It means that the statement P is true for the element c.

The overall statement is saying that if there is at least one element that satisfies the property P, then there exists an element c for which the property P is true.

In a mathematical context, this implication is often used in proofs or logical arguments where the existence of an element with a certain property implies the existence of another element with the same property. It recognizes that if you can find one example that satisfies a property, you can conclude that such an example exists for any specific chosen element c.

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