The Universal Quantifier Principle and Implication | If P(c) holds true for any value of c, then ∀xP(x)

P(c) for an arbitrary c___________∴∀xP(x)

To answer this question, we need to understand the notation and the meaning of the symbols used

To answer this question, we need to understand the notation and the meaning of the symbols used.

First, let’s define the symbols:
– P(c): This represents a proposition or statement involving the variable c. It could be any statement, such as “c is greater than 5” or “c is a prime number.”
– ∀x: This is the universal quantifier symbol, which means “for all” or “for every.” It is used to indicate that the statement following it applies to all elements in the set.
– P(x): This is similar to P(c), but instead of a specific variable c, it represents a proposition involving an arbitrary variable x.

Now, let’s interpret the given statement:

P(c) for an arbitrary c: This means that P(c) holds true for any value of c. In other words, no matter what value you substitute in place of c, the statement P(c) will be true.

∴: This symbol means “therefore” and is used to indicate a logical conclusion.

∀xP(x): This statement represents a universal proposition, stating that for every value of x, the statement P(x) holds true. It is saying that the proposition P(x) applies to all elements in the set.

Putting it all together, the given statement is stating that if P(c) is true for any value of c, then it implies that the proposition P(x) holds true for all values of x.

In other words, if a statement is true for any specific value (P(c)), then it must be true for all values (∀xP(x)). This is a generalization or inference based on the assumption that P(c) holds true for any specific value of c.

More Answers: