I am having a basic mathematics course on Number System in which my professor defined Intezers using equivalent relations on N and then finally he used a function to claim that the numbers in Natural number set are mathematically same as that in positive integers.
Now there were some sub parts of that function and I couldn't really interpret the meaning of each of those sub parts.
Now i understand the meaning of defining it to be one-one , so that one number in N is mapped to one number in Z but what's the importance of the fourth statement ? How do they help in embedding N in Z ?
To embed $\Bbb N\hookrightarrow\Bbb Z$ in the sense of sets you just need an injection. $f$ is an injection, this is bullet point $(i)$. From this perspective, the other bullet points are redundant.
However:
Bullet point $(ii)$ tells you we can view $\Bbb N\subset\Bbb Z$ in the nice and obvious way, as the integers $1,2,3,\cdots\in\Bbb Z$ (a priori it could be possible that there is no 'good' embedding $\Bbb N\hookrightarrow\Bbb Z$ with image $1,2,3,\cdots$ so this bullet point is not redundant!)
Bullet points $(iii)$ and $(iv)$ tell you that this embedding is 'good'. Preserving the identity $(1)$ and preserving addition and multiplication means we've preserved the 'structure' on $\Bbb N$, so $f:\Bbb N\hookrightarrow\Bbb Z$ is not just an embedding of sets but an embedding of 'structure'. You could make this precise by saying it $f$ is a morphism of rigs (rings without additive inverses). If you've studied any abstract algebra I'd hope you appreciate the importance of the concept of a (homo)morphism.
After all, you know $\Bbb Z$ contains $\Bbb N$, but you'd also demand that $1+1$ (as evaluated in $\Bbb Z$) equals $1+1$ (as evaluated in $\Bbb N$). Then, there is 'no difference' in viewing $\Bbb N$ on its own or viewing $\Bbb N$ as a subset/substructure/subobject of $\Bbb Z$. This is good! It is the basis of much of the primary school mathematics we take for granted.