Algebra

Question :

For each of the following, let the operation ∗ be defined on Z by the given rule.
Determine in each case whether Z is a group with respect to ∗ and whether it is an
abelian group. State which, if any, conditions fail to hold.
(a) x ∗ y = x + y + 1
(b) x ∗ y = x − y
(c) x ∗ y = x + xy + y

In the question 'Z' in here is given by the usual notation of the set of integers(Appears hear as Z due to format changes occurs in the site)

So in this type of a question,

1) can we assume Z denote the set of integers because it's given in the usual notation? And hence can we apply the properties of real numbers to prove that 'Z' here is a group, since the set of integers is a subset of the set of integers?

2) Can we assume the '+' given here denotes the usual operation of addition?

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  1. yes to both questions

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