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If $N=(0,1,2,3 ....)$ , then $(N,+)$ is a Group.

If $N=(1,2,3 ....)$ , then $(N,+)$ is a not Group.

Which one to consider in exam ?
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Whether or not '0' belongs to the set of natural numbers ℕ is a debatable topic. Although, the definition we've been taught makes a distinction between natural numbers and whole numbers via inclusion or exclusion of zero but still it's always good to mention what definition are we talking about.

So, in my opinion, any standard exam will specify the nomenclature it refers to. There can be workarounds like using ℕ+ = {1,2,3, ...} or ℕ0 = ℕ0 = {0, 1, 2, …}. 

PS: If I were you faced with such a situation, I would have marked it a group because according to Peano Axioms, (the first axiom) - "The constant zero is a natural number".

For more information on this refer the WIkipedia page. https://en.wikipedia.org/wiki/Natural_number

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