Why is it true that Max(X) = Min(-X) for a group X composed of numbers in Z I'm looking for a formal explanation that won't involve calculus (if possible) that would explain why for every group of numbers X (such that all numbers in X are integers), Max(X) = Min(-X) (where -X denotes changing all values in group X to - the values).
Any explanation would be greatly appreciated.
 A: What you are implicitly asking about is the property of the Least Upper Bound and the Greatest Lower Bound, and the relationship between the two. Though I won't bother you with the details in this answer, you should read up on them if you are interested.
Now to the formal answer of your question, as requested:
Proposition: For any $X \subseteq \mathbb{Z}$, $\max{(X)} = \min{(-X)}$, where $-X = \{x \in X \mid x \cdot (-1) \}$
Proof:
Take $x \in X$ s.t. $x = \max(X)$. Now take $-x \in -X$. Suppose $-x$ is not the minimal element of $-X$. Then there exists $-y \in Y$ s.t. $-x > -y$ and $-y = \min(-X)$. Now consider the elements of $X$ that correspond to $-y$ and $-x$: because we have $-y \in -X$, there must exist a $(-y \cdot (-1)) \in X$. Similar for $-x$, as we know. 
But if $-x > -y$, then $x < y$. But we know $x$ is the largest element of $X$, and $y>x$ implies that $y = \max(X)$, which is a contradiction. So our supposition that $-x$ is not the minimal element of $-X$ must be false. So by contradiction, $-x = \min(-X)$.
