Logic and Math | Understanding the Relationship between Propositions and Existence

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

In this question, the statement “P(c)” represents the proposition or condition that P is true for some element c

In this question, the statement “P(c)” represents the proposition or condition that P is true for some element c. The symbol “∴” means “therefore” or “thus.” The statement “∃xP(x)” represents the existence of an element x such that P(x) is true.

To break down the given statement:

P(c) for some element c: This means that the proposition P is true for at least one specific element c. In other words, there exists an element c for which P(c) is true.

∴: This symbol implies that the statement on the right-hand side follows or is a logical consequence of the statement on the left-hand side.

∃xP(x): This statement represents the existence of an element x for which P(x) is true. It is a way to express that there is at least one x that satisfies the condition P.

Putting it all together, the given statement can be read as follows:

If P is true for some element c, it follows that there exists an element x for which P(x) is true.

In other words, if there is at least one element c for which P is true, then we can conclude that there exists some element x for which P is true.

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