Symbolic form is $p \vee (\sim p \wedge \sim q) \vee q$
is equivalent to $p \vee q$
Show Solution
The Correct Option isA
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the correct logical interpretation of a switching circuit containing four switches $S_1, S_1', S_2, S_2'$. Step 3: Detailed Explanation:
1. Identify the switches from the diagram: $S_1$ is represented by $p$, $S_2$ by $q$, $S_1'$ by $\sim p$, and $S_2'$ by $\sim q$.
2. The circuit shows three parallel branches connected to the lamp.
3. Branch 1 contains switch $S_1$. Symbolic representation: $p$.
4. Branch 2 contains switches $S_1'$ and $S_2'$ in series. Symbolic representation: $(\sim p \wedge \sim q)$.
5. Branch 3 contains switch $S_2$. Symbolic representation: $q$.
6. Since the branches are in parallel, we use the OR ($\vee$) operator between them.
7. The total symbolic form is $p \vee (\sim p \wedge \sim q) \vee q$.
8. Let's simplify this:
$p \vee (\sim p \wedge \sim q) \vee q \equiv (p \vee \sim p) \wedge (p \vee \sim q) \vee q$
$\equiv T \wedge (p \vee \sim q) \vee q$
$\equiv p \vee \sim q \vee q \equiv p \vee (\sim q \vee q) \equiv p \vee T \equiv T$.
Since it simplifies to $T$, the lamp is always on (Option A is also logically correct, but C gives the specific required interpretation). Step 4: Final Answer:
The symbolic form is $p \vee (\sim p \wedge \sim q) \vee q$.
