Question

If $p \equiv$ The switch $S_1$ is closed, $q \equiv$ The switch $S_2$ is closed, $r \equiv$ switch $S_3$ is closed, then symbolic form of the switching circuit is equivalent to \dots

• A. $p$
• B. $q$
• C. $p \wedge q$
• D. $p \vee q$

Hint:

Always look for the Absorption Law ($p \vee (p \wedge q) = p$ and $p \wedge (p \vee q) = p$) when dealing with logic circuit simplifications, as it is the most common way multiple switches collapse into a single variable.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to evaluate the simplified logical equivalent of a specific electrical switching circuit involving switches $S_1$, $S_2$, and $S_3$ (represented by statements $p$, $q$, and $r$).
Note: Since the visual diagram of the circuit was not provided in the text transcript, we will detail the general methodology used to solve such problems, assuming the circuit is a standard arrangement that perfectly simplifies to $p$.

Step 2: Key Formula or Approach:
1. Series Connection: If switches are in series (one after another), they represent the logical "AND" operation ($\wedge$).
2. Parallel Connection: If switches are in parallel (stacked on separate branches), they represent the logical "OR" operation ($\vee$).
3. Simplification: Apply Boolean algebra laws (Distributive, Absorption, Identity) to minimize the expression.

Step 3: Detailed Explanation (Example Scenario):
Let's assume the missing diagram showed switch $S_1$ ($p$) in series with a parallel block containing $S_2$ ($q$) and another $S_1$ ($p$).
The initial logical expression would be: $p \wedge (q \vee p)$.
Using the Absorption Law of logic: $A \wedge (A \vee B) \equiv A$.
Applying this to our expression: $p \wedge (p \vee q) \equiv p$.
Thus, the entire complex circuit is logically equivalent to just having a single switch $S_1$ ($p$). The state of $q$ (and $r$) does not affect the final output.

Step 4: Final Answer:
Assuming the standard MHT CET reduction problem, the answer simplifies to $p$, matching option (a).