| 释义 |
quantifier The two expressions ‘for all…' and ‘there exists…' are called quantifiers. A phrase such as ‘for all x' or ‘there exists x' may stand in front of a sentence involving a symbol x and thereby create a statement that makes sense and is either true or false. There are different ways in English of expressing the same sense as ‘for all x', but it is sometimes useful to standardize the language to this particular form. This is known as a universal quantifier and is written in symbols as ‘∀x'. Similarly, ‘there exists x' may be used as the standard form to replace any phrase with this meaning and is an existential quantifier, written in symbols as ‘∃x'.
For example, the statements ‘if x is any number greater than 3 then x is positive' and ‘there is a real number satisfying x2=2' can be written in more standard form: ‘for all x, if x is greater than 3 then x is positive', and ‘there exists x such that x is real and x2=2'. These can be written, using the symbols of mathematical logic, as: (∀x)(x>3 ⇒ x>0), and (∃x)(x ϵ R ∧ x2=2). |