(317) Statements with/out variables

Here we will sum up statements with or without variables.

If P is a statement without variables, it has truth values. There are only two truth values: true or false.

On the other hand, P(x) is a statement with variable x. It has a truth set, where each x_0 is the set is a value that makes P(x_0) true. It is possible that all x in the universe makes P(x) true; or there are only some values that make it true.

For example:

P: ” 3 is a prime number.” is  a statement, and there is no variable. The truth value of P depends on itself and nothing else.

P(x): “x is a prime number.” is a statement, and x is the variable. The truth set of P(x) is the set of all prime numbers, which is a subset of \mathbb Z.

Quantifiers bind variables, and they only make sense for statements with variables. Both \forall xP(x) and \exists xP(x) have no free variable, and therefore become statements. They now have only truth values true and false.

As \forall xP(x) and \exists xP(x) are really the shorthand notation for \forall x\in UP(x) and \exists x\in UP(x), the truth value would depend on both P(x) and the universe U.

In our previous example ofP(x), adding the universal quantifier makes the statement into \forall xP(x), which means “all x are prime.” This statement is either true or false. It is true, if the universe we choose is the set of all prime numbers, and false if the universe is all integers \mathbb Z.

Similarly, the existential quantifier makes the statement into \exists xP(x), which means “there are some x that is prime.” This statement is either true or false. If the intersection of the universe and the set of prime numbers is non-empty, for example U=\{4.5, 5, \pi, e,18\}, then the statement is true. If the universe doesn’t contain any prime numbers, for example U=\{4,6,8,18\}, then the statement is false.

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