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

[ozkan12]

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 ?

 

[Evgeny.Makarov]

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.

 

[ozkan12]

that is, for f:X >>>X if f(X)CX then f is selfmapping ? is it true, I don't understand...

 

[Evgeny.Makarov]

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.

 

[ozkan12]

I understand but if f(X)⊆X I say f is invariant map...is it true ? I asked this ?

 

[Evgeny.Makarov]

I am not familiar with the term "invariant map".

 

[ozkan12]

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

 

[Evgeny.Makarov]

İf f(X)⊆X then f is selfmaping ?

What is $X$ here?

 

[ozkan12]

X is non empty set and X is metric space

 

[Evgeny.Makarov]

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.