Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990
Association of Mizar Users
The Modification of a Function by a Function
and the Iteration of the Composition of a Function

Czeslaw Bylinski

Warsaw University, Bialystok

Supported by RPBP.III24.C1.
Summary.

In the article we introduce some operations on functions.
We define the natural ordering relation on functions. The fact that
a function $f$ is less than a function $g$ we denote by $f \leq g$
and we define by $\hbox{graph} f \subseteq \hbox{graph} f$.
In the sequel we define the modifications of a function $f$ by a function $g$
denoted $f \hbox{+$\cdot$} g$ and the $n$th iteration of the composition of
a function $f$ denoted by $f^n$.
We prove some propositions related to the introduced notions.
MML Identifier:
FUNCT_4
The terminology and notation used in this paper have been
introduced in the following articles
[6]
[4]
[7]
[8]
[1]
[9]
[2]
[3]
[5]
Contents (PDF format)
Bibliography
 [1]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
 [2]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
 [3]
Czeslaw Bylinski.
Partial functions.
Journal of Formalized Mathematics,
1, 1989.
 [4]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
 [5]
Andrzej Trybulec.
Binary operations applied to functions.
Journal of Formalized Mathematics,
1, 1989.
 [6]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
 [7]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
 [8]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
 [9]
Edmund Woronowicz.
Relations defined on sets.
Journal of Formalized Mathematics,
1, 1989.
Received March 1, 1990
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