What is the relationship between $f$ and $X$?

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    Invariant Map Self
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Discussion Overview

The discussion revolves around the relationship between a function \( f \) and a set \( X \), specifically focusing on the concepts of self-mapping and invariant sets within the context of functions defined from a set to itself. The scope includes theoretical aspects of function definitions and properties in mathematics.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that if \( f: X \to X \) and \( f(X) \subseteq X \), then \( f \) is invariant and questions if this implies \( f \) is a self-mapping.
  • Another participant clarifies that the notation \( f: X \to X \) indicates both the domain and codomain are \( X \), and that \( f(X) \subseteq X \) is a necessary condition for defining invariant sets.
  • A participant expresses confusion about the term "self-mapping" and seeks clarification on whether \( f(X) \subseteq X \) implies \( f \) is a self-mapping.
  • It is noted that the condition \( f(X) \subseteq X \) is considered superfluous by one participant, as a function defined from \( X \) to \( X \) inherently satisfies this condition.
  • Several participants express uncertainty about the definition of an "invariant map" and its relationship to self-mapping.
  • One participant asserts that \( f(X) \subseteq X \) does not necessarily imply that \( f \) is a self-mapping, emphasizing the distinction between the two concepts.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the implications of \( f(X) \subseteq X \) on whether \( f \) can be classified as a self-mapping. Some participants assert that this condition does not suffice to establish \( f \) as a self-mapping, while others question the definitions and seek clarification.

Contextual Notes

There are unresolved definitions and terminological differences regarding "invariant map" and "self-mapping," which contribute to the confusion in the discussion. The relationship between the properties of \( f \) and the set \( X \) remains a point of contention.

Who May Find This Useful

This discussion may be of interest to those studying mathematical functions, particularly in the context of set theory and functional analysis, as well as individuals exploring the nuances of mathematical terminology.

ozkan12
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let f:X to X be and f(X) C X...then f is invariant..if f is invariant, then f is self map on X ? is it true ?
 
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The notation $f:X\to X$ means that both the domain and the codomain of $f$ are $X$. By definition, the image of the domain is a subset of the codomain, i.e., $f(X)\subseteq X$.

As for terminology, there are apparently difficulties arising from translation. A function whose domain and codomain coincide is called an endofunction or self-mapping, though these are not very popular terms. If $f:X\to X$ and $A\subseteq X$, then $A$ is called an invariant set if $f(A)\subseteq A$. An example is an invariant subspace of a linear vector space.
 
that is, for f:X >>>X if f(X)CX then f is selfmapping ? is it true, I don't understand...
 
ozkan12 said:
that is, for f:X >>>X if f(X)CX then f is selfmapping ?
The condition $f(X)\subseteq X$ is superfluous. A function $f$ is called a self-mapping if $f:X\to X$. In this case, $f(X)\subseteq X$ holds automatically.

I recommend you start learning LaTeX by clicking on the "Reply With Quote" button under a post and examining how others type mathematical formulas.
 
I understand but if f(X)⊆X I say f is invariant map...is it true ? I asked this ?
 
I am not familiar with the term "invariant map".
 
ok I understand.. İf f: X to X then f(X)⊆X..I know this...but İf f(X)⊆X then f is selfmaping ? is it true ? I really wonder this
 
ozkan12 said:
İf f(X)⊆X then f is selfmaping ?
What is $X$ here?
 
X is non empty set and X is metric space
 
  • #10
I meant, what is the relationship between $f$ and $X$?

In general, no, $f(X)\subseteq X$ does not mean that $f$ is a self-mapping. The latter means that both the domain and the codomain of $f$ are $X$. At least, according to the link I gave.
 

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