$P, Q$ and $R$ represent switches in on position and $P^{'}, Q^{'}$ and $R^{'}$ represent switches in off position. Construct a switching circuit representing the polynomial $P R + Q (Q^{'} + R) (P + Q R)$ . Using Boolean Algebra, simplify the polynomial expression and construct the simplified circuit

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Solution

The switching circuit of the given polynomial is
The given polynomial can be simplified as:
$P R + Q (Q^{'} + R) (P + Q R) = P R + (Q Q^{'} + Q R) (P + Q R)$
$= P R + (0 + Q R) (P + Q R) (∵ Q Q^{'} = 0)$
$= P R + Q R (P + Q R)$
$= P R + Q R [∵ a (a + b) = a]$
$\Rightarrow$ The light is on when the switches $P$ and $R$ are on or when the switches $Q$ and $R$ are on.
Equivalent switching circuit is shown in fgure

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