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:

Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is

Determine whether each of the following is a tautology, a contradiction, or neither ("$$ \vee $$" is disjunction, "$$ \wedge $$" is conjuction, "$$ \to $$" is implication, "$$\neg $$" is negation, and "$$ \leftrightarrow $$" is biconditional (if and only if).

(i)$$\,\,\,\,\,\,A \leftrightarrow \left( {A \vee A} \right)$$
(ii)$$\,\,\,\,\,\,\left( {A \vee B} \right) \to B$$
(iii)$$\,\,\,\,\,\,A \vee \left( {\neg \left( {A \vee B} \right)} \right)$$

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 $${N_f}$$?