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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?

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4 Answers

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No it isn't the identity since it isn't surjective. In the case when $B=A$ then this injection is the identity.

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Recall the definition: Two functions are equal if they have the same domain, codomain and if for each $x$ in the domain they take the same value...

You are missing the bolded part.

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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.

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Two maps $f,g$ are the same if their domain is the same and $f(x)=g(x) \hspace{0.5cm}\forall \hspace{0.2cm}x \in D=D(f)=D(g)$. In your case the first condition is not satisfied.

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