Question

The following are attempts to define a binary operation on a set, are they actually binary...

The following are attempts to define a binary operation on a set, are they actually binary operations
on the given set? If yes, prove it and if not please provide an explanation.

1) a*b = a-b on S, S is the set Z of integers.

2) a*b = a log b on S, S is the set R+ of positive real numbers

3) a*b = |a+b| on S, S is the set of Real numbers.

what I want to know about each of them first is:

1) what is the question asking? is it to prove the three axioms ( Identity, Inverse, Associative) that was defined when we were learning Binary operations, or is it just prove that for each value in the given set, the result of the right handside of the equation is still in the given set?

2) if it is to prove the three axioms, then how to prove it. I am so cnfused, thanks if anyone can help!

Math   Algebra   
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