Question

If p : switch $S_1$ is closed, q : switch $S_2$ is closed then correct interpretation from the following circuit is

• A. The lamp is always on
• B. The lamp is always off
• C. Symbolic form is $p \vee (\sim p \wedge \sim q) \vee q$
• D. is equivalent to $p \vee q$

Hint:

Parallel connections use $\vee$ (OR); Series connections use $\wedge$ (AND).

Answer 1

The Correct Option is A

Step 1: Write Symbolic Form
The circuit consists of three parallel branches:
1. Switch $S_1$ ($p$)
2. Switches $S_1'$ and $S_2'$ in series ($\sim p \wedge \sim q$)
3. Switch $S_2$ ($q$)
Symbolic form: $L \equiv p \vee (\sim p \wedge \sim q) \vee q$.

Step 2: Simplify Using Laws of Logic
$L \equiv (p \vee q) \vee (\sim p \wedge \sim q)$
$L \equiv (p \vee q) \vee \sim(p \vee q)$ (By De Morgan's Law).

Step 3: Conclusion
This is of the form $X \vee \sim X$, which is a tautology ($T$).
Since the expression is always true, the lamp is always on.
Final Answer: (A)