TITLE: Is inclusion map not the same as identity?
QUESTION [19 upvotes]: Wikipedia says $i$ is an inclusion means $i: A \to B$ with $A \subset B$ means $i(x) = x$ for each $x\in A$. 
But doesn't this mean $i(A) = A$, so this is actually identity?

REPLY [2 votes]: Map i is not identity map,to be identity codomain and domain has to be same.
Also, we can say map i is identity on A.