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

(a) $$S = \left\{ { < 1,2 > ,\, < 2,1 > } \right\}$$ is binary relation on set $$A = \left\{ {1,2,3} \right\}$$. Is it irreflexive?
Add the minimumnumber of ordered pairs to $$S$$ to make it an $$\,\,\,\,\,$$equivalence relation. Give the modified $$S$$.

(b) Let $$S = \left\{ {a,\,\,b} \right\}\,\,\,\,$$ and let ▢ $$S$$ be the power set of $$S$$. Consider the binary relation $$'\underline \subset $$ (set inclusion)' on ▢ $$S$$. Draw the Hasse diagram corresponding to the lattice (▢$$(S)$$, $$\underline \subset $$)