## ∀x P(x)

### The notation ∀x P(x) represents a mathematical statement called a universal quantification

The notation ∀x P(x) represents a mathematical statement called a universal quantification. It is read as “for all x, P(x)” or “for every x, P(x)”.

In mathematics, it is common to use variables, such as x, to represent elements or values from a given set. The statement ∀x P(x) asserts that P(x) is true for every possible value of x in the given set.

To understand this in more detail, let’s break it down:

– The symbol ∀ (read as “for all” or “for every”) indicates that the subsequent statement applies to all elements in a set.

– The variable x represents an arbitrary element from the set under consideration.

– P(x) is a predicate or proposition that depends on the variable x. It represents a statement that may or may not be true for a specific value of x.

By stating ∀x P(x), we are claiming that the proposition P(x) is true for every possible value of x in the given set. This universal quantification allows us to make general statements about the entire set of elements without explicitly specifying each element.

For example, if we have a set of integers, and P(x) represents the statement “x is an even number,” then ∀x P(x) asserts that every integer is an even number. This statement would be false since there are odd integers that do not satisfy the property P(x).

Overall, the notation ∀x P(x) is a concise and powerful way to express statements that apply to all elements in a set and is commonly used in mathematical logic and formal proofs.

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