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

The binary relation $$S = \phi $$ (emply set) on set A = {1, 2, 3} 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)$$\,\,\,\,\,\,A \vee \left( {\neg \left( {A \vee B} \right)} \right)$$

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: