Possible binary operation is $a*b = a – b + 1$ for all $a,b \in \mathbb{Z}^+$
I don't believe this is a binary operation in any way. A counter example is when $a = 1$ and $b = 5$, then the result of $a*b, -4 \notin \mathbb{Z}^+$
I also checked whether it was commutative or associative. It failed both tests. However, I don't think I actually needed to check for those properties.
A binary operation can be neither associative or commutative, but still be considered a binary operation I suppose, so all that is necessary to be a binary operation is if the result is in the same set?
Best Answer
It is not a binary operation because a binary operation on a set $X$ is a mapping (any mapping) from $X\times X$ to $X$. As you correctly pointed out, your mapping does not mapp from $\mathbb Z_+\times \mathbb Z_+$ to $\mathbb Z_+$, because it maps $(1,5)$ to $-4$ which is not in $\mathbb Z_+$.
Checking for commutativity and associativity, however, is meaningless in the context of the question "is this a binary operation?". There exist binary operations which are neither commutative nor associative, but they are still binary operations. Testing for commutativity has no effect on the question you are trying to answer.