I'm having trouble with this question on my discrete math HW:
Let R be an equivalence relation defined on a finite set A, such that |A| = n and |R|= m. Let k be the number of equivalence classes of R.
Prove that $n^2 \le km$
The professor gave the hit to use the Cauchy-Shwarz Inequality:
$(\sum_{i=1}^px_iy_i)^2 \le (\sum_{i=1}^px_i^2)(\sum_{i=1}^py_i^2)$
My initial thought process was to try to represent |A| as $\sum_{i=1}^px_iy_i$, |R| as $\sum_{i=1}^px_i^2$ and k or the # of equivalence classes as $\sum_{i=1}^py_i^2$ but I'm having trouble with that.