Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

**Katarzyna Jankowska**- Warsaw University, Bialystok

- Some facts concerning matrices with dimension $2\times 2$ are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. Some proofs are based on [11] and [14].

- Some examples of matrices
- Deleting of rows and columns in a matrix
- Sets of permutations
- Group of permutations

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Binary operations.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [11] Thomas W. Hungerford. \em Algebra, volume 73 of \em Graduate Texts in Mathematics. Springer-Verlag New York Inc., Seattle, Washington USA, Department of Mathematics University of Washington edition, 1974.
- [12]
Katarzyna Jankowska.
Matrices. Abelian group of matrices.
*Journal of Formalized Mathematics*, 3, 1991. - [13]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [14] Serge Lang. \em Algebra. PWN, Warszawa, 1984.
- [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [16]
Andrzej Trybulec and Agata Darmochwal.
Boolean domains.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [19]
Wojciech A. Trybulec.
Lattice of subgroups of a group. Frattini subgroup.
*Journal of Formalized Mathematics*, 2, 1990. - [20]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

[ Download a postscript version, MML identifier index, Mizar home page]