Give an alternative equivalent simple circuit for the following switching circuit :[Figure: Switch \(S_1\) is connected in series with a parallel combination of two switches \(S_1'\) (upper branch) and \(S_2\) (lower branch); the circuit is completed…

Answer: An equivalent simple circuit has the two switches \(S_1\) and \(S_2\) in series with the lamp, i.e. \(p \wedge q\).

Let \(p\): switch \(S_1\) is closed, \(q\): switch \(S_2\) is closed. Then \(\sim p\): switch \(S_1'\) is closed (i.e. \(S_1\) is open).

The symbolic form of the given circuit is \(p \wedge (\sim p \vee q)\).

\(p \wedge (\sim p \vee q) \equiv (p \wedge \sim p) \vee (p \wedge q)\) (distributive law)

\(\equiv F \vee (p \wedge q)\) (complement law)

\(\equiv p \wedge q\) (identity law)

Check by truth table:

[Figure: Alternative simple circuit: switches \(S_1\) and \(S_2\) in series with the battery and the lamp L.]