Discrete math question... Finding largest area Can anyone help explain to me how to go about solving this question ?:
Of all the rectangles with perimeter 4, prove that the square has the largest area.
I know how to solve this algebraically, although how would one go about solving it in context of discrete mathematics 

Let $n \in \mathbb{N}$. Imagine a complete bipartite graph with a total of $2n$ vertices. Let $p + q = 2n$ be the number of vertices on left and right sides of the graph.
This graph has $p\cdot q$ vertices. Imagine a transformation, where you take one vertex from left side and move it to the right side, i.e. $p' = p-1$ and $q' = q+1$. Observe, that this operation removes $q$ edges (we take a vertex from the left) and adds $p-1$ edges (we add vertex to the right). Of course, it is also possible to calculate $(p-1)\cdot(q+1) - p\cdot q = p-q-1$. This tells us, that as long as $p > q+1$ we could transform the graph in to one with more edges.
Finally we arrive, that $p > q+1 \implies |E|<|E'|$ and by symmetry when $q > p+1 \implies |E|>|E'|$. Given that $p+q = 2n$ we conclude that $|E|$ is maximal for $p = q = n$.
I hope it is what you were asking for (put $n = 4$).   
