A certain system requires that all three switches $1,2,3,$

must be turned on for the system to operate. Consider the events

$$ A: switch $1$ fails to turn on;

$$ B: switch $2$ fails to turn on;

$$ C: switch $3$ fails to turn on$.$

Express the following events in terms of unions, intersections, and complements of the events A$,$ B$,$ and C$.$ No justification is needed for

this question.

$($a$)$ Switch $3$ fails to turns on$,$ and at least one of switch $1$ or switch

$2$ fails to turn on$.$

$($b$)$ Switch $1$ fails to turn on$,$ while both switch $2$ and switch $3$ are

turned on$.($c$)$ Switch $1$ is turned on$,$ switch $2$ fails to turn on$,$ and switch $3$ is

turned on$.$

$($d$)$ All three switches are turned on