Showing why $\Bbb Z^-∩\Bbb Z^+=∅$ is true 
My Problem: Is this statement true or false? Give a reason. If false a counter example is adequate.
  $$\Bbb Z^-∩\Bbb Z^+=∅$$

I'm more so after the reasoning as I don't know how to word it. I know the statement is true.
 A: I don't really think there is any particularly "elegant" logic needed - simply that they don't share a common element. It basically amounts to "there is no integer which is both positive and negative."
I guess if you want to argue it in a somewhat formal way, you can say that, for all $n \in \Bbb Z^-$, $n < 0$, and for all $m \in \Bbb Z^+$, $m > 0$. If there exists an element $k$ in the intersection of the two, this means $k<0$, but also $k>0$, which is simply not possible.
A: An intersection of two sets is the set of elements that are in both sets.
$\mathbb Z^+$ is the set of integers that are positive and $\mathbb Z^-$ is the set of integer that are negative.  So $\mathbb Z^+ \cap \mathbb Z^+$ are the set of integers that are both positive and negative.
To be negative is to be less than $0$ and to be positive is to be bigger than $0$. A fundamental principal of real numbers is that for any $a,b$ either $a < b; a=b$ or $a > b$-- exactly one and only one of those is true.  So an integer being both greater than and less than $0$ is impossible.
