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?
yes to both questions
1) Yes, in this case, we can assume that Z denotes the set of integers because it is given in the usual notation. Since the set of integers is a subset of the set of real numbers, we can apply the properties of real numbers to prove whether Z is a group.
2) Yes, we can assume the '+' given here denotes the usual operation of addition. In mathematics, the symbol '+' typically represents addition and is usually defined as the operation of combining two numbers to produce their sum. So, in this case, we can interpret the '+' as ordinary addition.