Quantification Sentence In Math Logic

An open sentence can be converted into a statement if the variable of the sentence is substituted by a certain constant.

For example:
Open sentence: x + 4 = 3 for x ∈ R
If x = -1, then the above sentence becomes a true value statement.
If x = 2, then the open sentence becomes a statement of false value.

Another way to turn an open sentence into a statement is to use a Quantification

There are two types of quantification in mathematical logic, such as:

1. Universal Quantification
2. Existential Quantification

1. Universal Quantification

Universal Quantification is written with the symbol "∀" and read "for all" or "for each". If p(x) is an open sentence and given a universal quantification it will be a revelation and written (∀x) p(x) read:

• For each price x apply the nature p.
• For all prices x has properties p.

The form (∀x) p(x) is a declarative statement that has a truth value to be true or false, ie if it can not be found x that is not p(x) then (∀x) p(x) is true. If x can be found that is not p(x), then (∀x) p(x) is false.

Example of Universal Quantification

∀x real number, x² = 1
This statement is false even if it applies to x = - 1 or x = 1 ie 1² = 1 and (1)² = 1 but does not apply to all x (eg x = 3, then 3² ≠ 1)

2. Existential Quantification

Existential Quantification is written with the symbol "∃" and read "any / some" or "at least one". If p (x) is an open sentence and given an existential query it will be a statement and written (∃x) p(x) read:

• there is x such that the applied properties p.
• some x have properties p.
• at least one x by nature p.

The form (∃x) p(x) is a declarative statement that has a truth value to be true or false ie if it can be found at least one x which is p(x) then (∃x) p(x) is true. If no x can be found in p(x) then (∃x) p(x) is false.

Example of Existential Quantification

∃x real number, x < 1
The statement is false because it can not be determined x the original number < 1.

Similarly this article.
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Referensi :

• To'Ali's book math group accounting and sales

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