# What is calculus in mathematics

## logical calculation

Quadruple \ ({\ mathcal {K}} \ text {= (} E, \, A, \, S, \, F \ text {)} \) with the following determinants:

*E.*is a non-empty set, the elements of which serve as the basic symbols of the calculus (*E.*is also called the alphabet of \ ({\ mathcal {K}} \)).*A.*is a suitable subset of the free semigroup*E.*^{*}above*E.*(*E.*^{*}is the set of all words, i.e. the finite strings of basic characters in \ ({\ mathcal {K}} \)).*A.*is called the set of expressions of \ ({\ mathcal {K}} \);*A.*is usually defined inductively (see also Propositional calculus, Predicate calculus).*S.*is a special set of expressions, the set of sentences of \ ({\ mathcal {K}} \).*F.*is a mapping (it is called the derivative relation of \ ({\ mathcal {K}} \)), that of each subset*X*⊆*A.*a part*F.*(*X*) ⊆*A.*with the following properties:*X*⊆*F.*(*X*),- if
*X*_{1}⊆*X*_{2}so*F.*(*X*) ⊆*F.*(*X*_{2}), *F.*(*F.*(*X*)) ⊆*F.*(*X*),- to each
*a*∈*F.*(*X*) there is a finite subset*X*_{0}⊆*X*, so that*a*∈*F.*(*X*_{0}), *F.*(*S.*) ⊆*S.*(⇒*F.*(*S.*) =*S.*).

*F.*, (d) symbolizes the finiteness law regarding*F.*, and (e) means that only (valid) sentences can be derived from (valid) sentences.

The most important examples of logical calculi are the propositional and predicate calculus (see also elementary language). A distinction is often made between calculi with a syntactically defined set of sentences (and the corresponding derivation relation) and semantically defined set of sentences (and a corresponding inference relation).

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