For a group we have four axioms: 1) Closure 2) Identity 3) Inverse and 4) Associativity. Can you show these?
Here is the question:
"Let G= Z x Q (Where Z is the set of integers and Q is the set of rational numbers) and define a binary operation o by
(a,b) o (c,d) = (a+c,2-cb+d) - (*)
(i) a semigroup,
(ii) a group?
You must give full reasoning for your answer in order to get full marks."
I require a natural proof to this and I am still a little new to this high level of maths so please excuse me for obvious errors.
I believe that I understand what the terms in the equation mean e.g. (binary operation - two elements with a calculation applied will produce an element in the same
(semigroup - a set with associated binary operation, in this case o, that meets the condition a o (b o c) = (a o b) o c
(Group - a binary operation attatched to a set)
I don't understand how we are to apply the above principles to equation (*).
Any help to shed some light on this would be greatly appreciated.