Show that proposition $$C$$ is a logical consequence of the formula $$A \wedge \left( {A \to \left( {B \vee C} \right) \wedge \left( {B \to \sim A} \right)} \right)$$ using truth tables.

Let $$S$$ be an infinite set and $${S_1},\,\,{S_2},....\,\,{S_n}$$ be sets such that $${S_1} \cup {S_2} \cup ....... \cup {S_n} = S$$. Then

Let $${\rm A}$$ be a finite set of size $$n$$. The number of elements in the power set of $${\rm A} \times {\rm A}$$ is

The eigen vector (s) of the matrix
$$\left[ {\matrix{ 0 & 0 & \alpha \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right],\alpha \ne 0$$ is (are)