Alphabet (math)

The articles alphabet (computer science) and alphabet (mathematics) overlap thematically. Help me to better differentiate or merge the articles (→  instructions ) . To do this, take part in the relevant redundancy discussion . Please remove this module only after the redundancy has been completely processed and do not forget to include the relevant entry on the redundancy discussion page{{ Done | 1 = ~~~~}}to mark. Ziegenberg ( discussion ) 8:28 p.m., Feb. 22, 2017 (CET)

An alphabet A is understood to be a non-empty set of characters or symbols. Alphabets are a central concept in theoretical computer science and are the basic building blocks of words, which in turn form the building blocks of languages. The central component of a logic is its underlying language. The alphabet of this language then indicates the number of permitted characters that can be used to build the terms and expressions of this logic. Finite linear rows of characters of an alphabet called character strings or words over A . The set of words is denoted by A ^* . Even the string that does not contain symbols is a word - the empty word. It is denoted by. ${\displaystyle \varepsilon }$

The alphabet of a first-order predicate logic

The alphabet of first-order predicate logic consists of the following characters:

1.   ${\displaystyle v_{0},v_{1},v_{2},\ldots \ -}$  Variable identifier ;
2.   ${\displaystyle \neg ,\wedge ,\vee ,\rightarrow ,\leftrightarrow \ -}$  Connections : negation (not), conjunction (and), disjunction (or), implication (if - then), equivalence (if and only if);
3.   ${\displaystyle \forall ,\exists \ -}$  universal and existential quantifier ;
4.   ${\displaystyle \equiv \ -}$  Equal sign;
5.   ${\displaystyle ),(,{\boldsymbol {,}}\ -}$  technical symbols: brackets and commas;
6.  In addition

a) a (possibly empty) set of constant symbols;

b) for each n 0 a (possibly empty) set of n -place relation symbols; ${\displaystyle \geq }$

c) for each n 0 a (possibly empty) set of n -place function symbols. ${\displaystyle \geq }$

Let A be the set of symbols listed in (1) to (5) and S the symbols from (6). Then call A _S the union of A and S . It's called A _S the alphabet of the first-order logic and S its symbol set. To specify the alphabet of a first-order predicate logic, it is sufficient to specify its symbol set, since the set A is the same for all alphabets for first-order predicate logics.

Sometimes you don't use the equal sign in an alphabet. In this case the symbol set must contain at least one relation symbol, otherwise no formulas can be created.

example

The most important first-level predicate logic , the language of set theory, contains only a single character in the set of symbols in its alphabet, namely the two-digit relational symbol (see set (mathematics) ). ${\displaystyle \in }$

Individual evidence

1. ↑ Hans Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic. Spektrum Akademischer Verlag, Heidelberg 2007, ISBN 3-8274-1691-4 .