# Is it a binary operation?

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?

• But the site said that i can't upload image because my reputation is below 10. @Joel_Reyes_Noche – Gurbir Singh Mar 22 '18 at 10:24
• I have added it. – Gurbir Singh Mar 22 '18 at 10:34
• are you claiming that 3[odd natural number]*2[even natural number]=6[even natural number] isn't a natural number? – nonchip Mar 22 '18 at 10:44
• @nonchip, I think the OP is claiming that $(3+2)/2=5/2$ is not a natural number. Note that the operation $*$ is not multiplication. – Joel Reyes Noche Mar 22 '18 at 10:47
• (3+2)÷2 is not a natural number. @nonchip – Gurbir Singh Mar 22 '18 at 10:47