Vtyjkyuk

I'm trying to wrap my head around inclusion maps and identity maps...

if X is a subset of Y, the function f defined by f(x)=x for each x in X is called the inclusion map of X into Y.

If X=1,2,3 and Y=1,2,3,4,5. Then f=(1,1),(2,2),(3,3)?

The inclusion map of X into X is called identity map on X. (In the language of relations, the identity map on X is the same as the relation of equality in X)

So in my example the inclusion map = identity map. Can someone give me an example of where this is not true?

Thanks in advance...

The inclusion map $Ato B$ is always the same set of pairs as the identity map on $A$.

So if you're in a context where maps are simply sets of pairs (and, in particular, a map has a range but does not have a unique codomain), then they are the same.

On the other hand, at least informally (and in quite a number of formal contexts too) it is usual and convenient to speak of a map as "knowing" what its domain and codomain are. In that case the inclusion map $Ato B$ differs from the identity map, because (assuming $B$ is a proper superset of $A$) their codomains are different -- even though both have the same domain and they have the same value at every point in the domain.

This perspective is needed, for example, if we want to ask whether the map is surjective or not. The identity map is a surjection; a nontrivial inclusion map is not.

A map is not defined only by a set of arrows $xmapsto f(x)$, it is defined by a domain and a codomain.

The maps $Bbb Rto Bbb R:xmapsto |x|$ and $Bbb Rto Bbb R_+:xmapsto |x|$ are two different maps.

If $Asubset B$,

• the inclusion map is defined as $Ato B : xmapsto x$,




• the identity map is defined as $Ato A : xmapsto x$.

The output is the same, but the codomain isn't.