Is it a binary operation?

Question

I am having problems with problem (i) shown in the image above.

Examine which of the following is a binary operation: (i) $a*b=\frac{a+b}{2}; a,b\in\mathbb{N}$

Now since $a$ and $b$ are natural numbers, they belong to the set $\mathbb{N}$. However the operation $a*b=(a+b)/2$ may not be a binary operation because for one (either $a$ or $b$) odd and the other even, the operation $a*b$ will not give a natural number. Hence the operation should not be a binary operation in $\mathbb{N}$. But the book has written that it is. So what is correct?

Answer 1

Your reasoning seems to be correct and the book is in error.