Step 1: Understanding the Concept:
Switching circuits can be represented by Boolean logic expressions.
Switches in series correspond to the logical AND ($\land$ or $\cdot$) operation.
Switches in parallel correspond to the logical OR ($\lor$ or $+$) operation.
Two circuits are equivalent if their Boolean logic expressions are logically equivalent. Step 2: Key Formula or Approach:
Write down the Boolean expression for each circuit diagram and simplify them using Boolean algebra laws, such as the distributive law: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$. Step 3: Detailed Explanation:
Let's analyze the given circuits: Circuit (A): Consists of two parallel branches. The top branch has switches $S_1$ and $S_2$ in series. The bottom branch has switches $S_1$ and $S_3$ in series.
The Boolean expression is: $L_A = (S_1 \land S_2) \lor (S_1 \land S_3)$. Circuit (B): Consists of two parallel branches. The top branch has switch $S_1$. The bottom branch has $S_2$ and $S_3$ in series.
The Boolean expression is: $L_B = S_1 \lor (S_2 \land S_3)$. Circuit (C): Consists of switch $S_1$ in series with a parallel combination of switches $S_2$ and $S_3$.
The Boolean expression is: $L_C = S_1 \land (S_2 \lor S_3)$.
By the distributive law of Boolean algebra:
$S_1 \land (S_2 \lor S_3) \equiv (S_1 \land S_2) \lor (S_1 \land S_3)$
This means the expression for Circuit (C) is logically equivalent to the expression for Circuit (A).
Therefore, Circuit (A) and Circuit (C) are equivalent.
Let's briefly look at the others to be sure:
Circuit (D) is $S_1 \land S_2 \land S_3$.
Circuit (E) has $S_1$ and $S_2$ in parallel, and then something else, but it's clear A and C are the intended pair demonstrating the distributive property. Step 4: Final Answer:
Circuits (A) and (C) are equivalent.
