Construct the switching circuit for the statement \((\sim p \wedge q) \vee (p \wedge \sim r)\).

Answer: Two branches in parallel: (\(S_1'\) in series with \(S_2\)) parallel to (\(S_1\) in series with \(S_3'\)), with the lamp and battery.

Let \(p\): switch \(S_1\) is closed, \(q\): switch \(S_2\) is closed, \(r\): switch \(S_3\) is closed. Then \(\sim p\) corresponds to \(S_1'\) and \(\sim r\) to \(S_3'\).

• \(\sim p \wedge q\): \(S_1'\) and \(S_2\) in series.
• \(p \wedge \sim r\): \(S_1\) and \(S_3'\) in series.
• The \(\vee\) between them: the two series branches are joined in parallel.

[Figure: a battery and a lamp L in series with a parallel combination of two branches; upper branch has \(S_1'\) and \(S_2\) in series, lower branch has \(S_1\) and \(S_3'\) in series.] The lamp glows exactly when \((\sim p \wedge q) \vee (p \wedge \sim r)\) is true.