Let X and Y be two exponentially distributed and independent random variables with mean $$\alpha $$ and $$\beta $$, respectively. If Z = min (X, Y), then the mean of Z is given by

Consider two events $${{E_1}}$$ and $${{E_2}}$$ such that probability of $${{E_1}}$$, Pr [$${{E_1}}$$] = 1/2, probability of $${{E_2}}$$, Pr[$${{E_2}}$$ = 1/3, and probability of $${{E_1}}$$ and $${{E_2}}$$, $$\left[ {{E_1}\,\,or\,\,{E_2}} \right]$$ = 1/5. Which of the following statements is /are true?

Let $$\left( {\left\{ {p,\,q} \right\},\, * } \right)$$ be a semi group where $$p * p = q$$. Show that: (a) $$p * q = q * p,$$, and (b) $$q * q = q$$

(a) Show that the formula $$\left[ {\left( { \sim p \vee Q} \right) \Rightarrow \left( {q \Rightarrow p} \right)} \right]$$ is not a tautology.

(b) Let $$A$$ be a tautology and $$B$$ be any other formula. Prove that $$\left( {A \vee B} \right)$$ is a tautology.