Let $*:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$ on the integers by the formula $x*y=x$ for any $x,y\in\mathbb{Z}$. Decide whether $*$ is associative and/or commutative.
To the best of my understanding this means I'm taking a Cartesian product $(x,y)$ and sending it to $x$. I think this would be both associative and commutative since it appears this only happens when the pair is $(x,0)$ with $(0,0)$.
Associativity means $(x*y)*z=x*(y*z)$.
By this, if $x=(x,0), y=(0,0)$, then $z$ must also be $(0,0)$ by necessity which clearly is associative.
Commutativity means $x*y=y*x$.
If $x=(x,0)$, then $y$ must be $(0,0)$. This is clearly commutative.
I'm struggling to put this into formal terminology. Are my thoughts correct? If not, what adjustments should I consider?
Best Answer
It is associative, because for all $x,y,z\in\mathbb{Z}$, you have that $$x\star (y\star z)=x\star y=x=(x\star y)\star z$$ It is not commutative, for example because $$0\star 1=0\neq 1=1\star 0$$