Question

Refer to question 6 above, state true or false (give reason for your answer)
(i) A and B are mutually exclusive
(ii) A and B are mutually exclusive and exhaustive
(iii) A = B'
(iv) A and C are mutually exclusive
(v) A and B' are mutually exclusive
(vi) A' B', C are mutually exclusive and exhaustive.

Answer 1

taking A, B, C events from question 6 Above we have
(i) $A\cap B=\varphi$
Thus A and B are mutually exclusive events.
$\therefore$ True.
(ii) $A\cap B=\varphi andA\cap B=S$
Thus A and B are mutually exclusive and exhaustive events.
$\therefore$ True.
(iii)$B^{\prime }=\left\{\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$
= A
$\therefore True$
$\left(iv\right)A\cap c=\left\{\right(2,1),(2,2),(2,3),(4,1\left)\right\}=\varphi$
Thus A and c are not mutually exclusive events $\therefore$ False
(v) $A\cap B^{\prime }=A\ne \varphi$
Thus A and B' are not mutually exclusive events.
(vi) Since $A^{\prime }=Band{B}^{\prime }=A,A\cap B=\varphi$
$B\cap C=\left\{\right(1,1),(1,2),(1,3),(1,4),(3,1),(3,2\left)\right\}$
$\ne \varphi$
$A\cap C=\left\{\right(2,1),(2,2),(2,3),(4,1\left)\right\}$
$=\varphi$
Thus A', B' and C are not mutually exclusive.
$\therefore$ False.