Q.Express the following switching circuit in symbolic form and simplify it: a branch with switches $p$ and $q$ in series, connected in parallel with a branch having switches $p$ and $\sim q$ in series.

Series is conjunction and parallel is disjunction; simplify the resulting form with the algebra of statements.

Symbolic form. Each series branch is a conjunction: the first branch is $p \wedge q$, the second is $p \wedge \sim q$. The two branches are in parallel, which is a disjunction, so the whole circuit is $(p \wedge q) \vee (p \wedge \sim q).$

Simplify. Take the common factor $p$ out by the distributive law: $(p \wedge q) \vee (p \wedge \sim q) \equiv p \wedge (q \vee \sim q).$ By the complement law $q \vee \sim q \equiv \mathbf{t}$ (always true), and by the identity law $p \wedge \mathbf{t} \equiv p$: $p \wedge (q \vee \sim q) \equiv p \wedge \mathbf{t} \equiv p.$

So the four-switch circuit is equivalent to a single switch $p$ — current flows exactly when $p$ is closed, regardless of $q$. …

One can also simplify by absorption after distribution, or verify directly by a truth table (shown a …

A common error is mixing up the connectives - using disjunction for a series connection or conjunction for a parallel one. Remember: series (both must be closed) = …