Give an alternative arrangement for the following circuit, so that new circuit has minimum switches :[Figure: Three branches connected in parallel between the two terminals, in series with a battery and a lamp L. Branch 1: switches \(S_1\) and \(S_2'…

Answer: Two switches \(S_1'\) and \(S_2'\) connected in parallel.

Let \(p\): switch \(S_1\) is closed, \(q\): switch \(S_2\) is closed; then \(\sim p\), \(\sim q\) represent \(S_1'\), \(S_2'\) closed.

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

Simplifying:

\((\sim p\wedge q)\vee(\sim p\wedge\sim q)\equiv\sim p\wedge(q\vee\sim q)\equiv\sim p\wedge T\equiv\sim p\) (distributive, complement and identity laws).

So the statement \(\equiv(p\wedge\sim q)\vee\sim p\equiv(p\vee\sim p)\wedge(\sim q\vee\sim p)\equiv T\wedge(\sim q\vee\sim p)\equiv\sim p\vee\sim q\).

Alternative circuit: switches \(S_1'\) and \(S_2'\) in parallel, in series with the battery and lamp L.

[Figure: switch \(S_1'\) and switch \(S_2'\) on two parallel branches, connected in series with the battery and the lamp L.]