## Binary operations applied to functions (1989)

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Venue: | Journal of Formalized Mathematics |

Citations: | 258 - 43 self |

### BibTeX

@ARTICLE{Trybulec89binaryoperations,

author = {Andrzej Trybulec},

title = {Binary operations applied to functions},

journal = {Journal of Formalized Mathematics},

year = {1989},

volume = {1}

}

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### Abstract

Summary. In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The recent concept is defined for an arbitrary function, however is meaningful in the case of functions which range is a subset of a Cartesian product of two sets. Then the converse of a function has the same domain as the function itself and assigns to an element of the domain the mirror image of the ordered pair assigned by the function. In the case of functions defined on a non-empty set we redefine the above mentioned functors and prove simplified versions of theorems proved in the general case. We prove also theorems stating relationships between introduced concepts and such properties of binary operations as commutativity or associativity.

### Citations

1278 | Tarski Grothendieck set theory
- Trybulec
- 1990
(Show Context)
Citation Context ...relationships between introduced concepts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles =-=[7]-=-, [6], [9], [8], [11], [10], [3], [5], [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B su... |

1257 | Functions and their basic properties
- Byliński
- 1990
(Show Context)
Citation Context ... concepts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], [8], [11], [10], =-=[3]-=-, [5], [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0 and B �= /0 and ... |

1187 | Properties of subsets
- Trybulec
- 1990
(Show Context)
Citation Context ...ips between introduced concepts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], =-=[9]-=-, [8], [11], [10], [3], [5], [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A ... |

1012 | Functions from a set to a set
- Byliński
- 1990
(Show Context)
Citation Context ...and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], [8], [11], [10], [3], [5], =-=[4]-=-, [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0 and B �= /0 and R = [:A, B... |

988 | Relations and their basic properties
- Woronowicz
- 1990
(Show Context)
Citation Context ...oduced concepts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], [8], [11], =-=[10]-=-, [3], [5], [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0 and B �= /0... |

725 | Some basic properties of sets
- Byliński
- 1990
(Show Context)
Citation Context ...ionships between introduced concepts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], =-=[6]-=-, [9], [8], [11], [10], [3], [5], [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such th... |

457 | Relations defined on sets
- Woronowicz
- 1990
(Show Context)
Citation Context ...n introduced concepts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], [8], =-=[11]-=-, [10], [3], [5], [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0 and B... |

432 | Partial functions
- Byliński
- 1990
(Show Context)
Citation Context ...epts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], [8], [11], [10], [3], =-=[5]-=-, [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0 and B �= /0 and R = [... |

331 | Binary operations
- Byliński
- 1990
(Show Context)
Citation Context ...rties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], [8], [11], [10], [3], [5], [4], [1], and =-=[2]-=- provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0 and B �= /0 and R = [:A, B:] holds domR ... |

269 | Tuples, projections and Cartesian products
- Trybulec
- 1990
(Show Context)
Citation Context ...etween introduced concepts and such properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], =-=[8]-=-, [11], [10], [3], [5], [4], [1], and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0... |

113 | Basic functions and operations on functions
- Byliński
- 1990
(Show Context)
Citation Context ...uch properties of binary operations as commutativity or associativity. MML Identifier:FUNCOP_1. WWW:http://mizar.org/JFM/Vol1/funcop_1.html The articles [7], [6], [9], [8], [11], [10], [3], [5], [4], =-=[1]-=-, and [2] provide the notation and terminology for this paper. The following proposition is true (1) For every binary relation R and for all sets A, B such that A �= /0 and B �= /0 and R = [:A, B:] ho... |