On (\mathbb{R}\setminus{0}), (a*b=\frac{ab}{a+b}), where (a+b\neq0). Why is it not a binary operation on the whole set?

Correct answer: Because the denominator becomes zero for some pairs

Step 1: A binary operation must be defined for every ordered pair of elements of the set. Step 2: If (a=1,b=-1), both are in the set, but (a+b=0), so the value is undefined. Step 3: One undefined pair is enough to reject it as a binary operation.