# Formal Languages - Concatenation and Closure

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## Formal Languages - Concatenation and Closure

Formal languages are introduced as subsets of the set of all 0-based finite sequences over a given set (the alphabet). Concatenation, the n-th power and closure are defined and their properties are shown. Finally, it is shown that the closure of the alphabet (understood here as the language of words of length 1) equals to the set of all words over that alphabet, and that the alphabet is the minimal set with this property. Notation and terminology were taken from [5] and [13].

MML identifier: FLANG 1, version: 7.8.04 4.81.962

Keywords:
References
• [3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

• [4] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

• [5] John E. Hopcroft and Jeffrey D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley Publishing Company, 1979.

• [6] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

• [7] Karol Pąak. The Catalan numbers. Part II. Formalized Mathematics, 14(4):153-159, 2006.

• [8] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

• [9] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.

• [10] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.

• [11] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

• [12] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.

• [13] William M. Waite and Gerhard Goos. Compiler Construction. Springer-Verlag New York Inc., 1984.

• [14] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

• [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

• [2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

# Formalized Mathematics

## (a computer assisted approach)

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