Question: Define a binary operation on the set of even integers which is different from addition,substraction and multiplication.
My attempt: $a \circ b = a + b - ab$ where $a\in G$ and $b\in G$ ($G$ is the set of all even numbers)
Is my answer is correct,i have mixed the addition, multiplication. Or $a \circ b = \max \left\{ a, b\right\}$ where $a\in G$ and $b\in G$ ($G$ is the set of all even numbers) .or this is correct?
Yes, both of them are correct since, in each case, if $a$ and $b$ are even integers, then $a\circ b$ is an even integer too.
A little bit more elaborated. Let the set $G$ be the set of all nubers divisible by $k$, so $$G=\{...-2k,-k,0,k,2k,3k,...\}$$ then $G$ is closed under operation you defined.
Take $a,b\in G$, then exist $x,y\in\mathbb{Z}$ so that $a=kx$ and $b=ky$. Now we have:
$$ a\circ b = kx+ky-kx\cdot ky = k(\underbrace{x+y-kxy}_{z\in \mathbb{Z}}) = kz$$ so $ a\circ b \in G$.
You where asking about the special case when $k=2$.
