From Wolf:

- A
*mathematical variable*is a symbol (or combination of symbols like ) that stands for an unspecified number or other object. The collections of objects from which any particular variable can take its values is called the*domain*or*universe*of that variable. Variables with the same domain are said to be of the same*sort*. - Two symbols, called
*quantifiers*, stand for the following words:- for ``for all'' or ``for every'' or ``for any''.
*Universal quantifier*. - for ``there exists'' or ``there is'' or ``for some''.
*Existential quantifier*.

- for ``for all'' or ``for every'' or ``for any''.
- The quantifiers are used in symbolic mathematical language as
follows: If
*P*is any statement, and*x*is any mathematical variable (not necessarily a real number variable), then and are also statements. In this case we say that*x*is*quantified*. - A mathematical variable occurring in a symbolic statement is
called
*free*if it is unquantified and*bound*if it is quantified. If a statement has no free variables it's called*closed*. Otherwise it's called a*predicate*, an*open sentence*, an*open statement*, or a*propositional function*. - A statement of the form is defined to be true
provided
*P*(*x*) is true for each particular value of*x*from its domain. Similarly, is defined to be true provided*P*(*x*) is true for at least one value of*x*from that domain. - For any statement
*P*(*x*), is logically equivalent to . Also, is logically equivalent to . - Let
*P*be any statement,*x*be any mathematical variable, and*t*any term that denotes a set. (So*t*could be a single letter standing for a set, or a more complicated expression like .) Then- is an abbreviation for .
- is an abbreviation for .

Wed Nov 18 12:16:44 EST 1998