The minimum number of colors required to color the vertices of a cycle with $$n$$ nodes in such a way that no two adjacent nodes have the same colour is:

The rank of the matrix$$\left[ {\matrix{ 1 & 1 \cr 0 & 0 \cr } } \right]\,\,is$$

Maximum number of edges in a n - node undirected graph without self loops is

Let $$A$$ be a set of $$n\left( { > 0} \right)$$ elements. Let $${N_r}$$ be the number of binary relations on $$A$$ and Let $${N_r}$$ be the number of functions from $$A$$ to $$A$$.
(a) Give the expression for $${N_r}$$ in terms of $$n$$.
(b) Give the expression for $${N_f}$$ in terms of $$n$$.
(c) Which is larger for all possible $$n, $$ $${N_r}$$ or $${N_f}$$?