Question:
The simplified switching circuit for the following circuit is
The simplified switching circuit for the following circuit is
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Write the Boolean expression of the circuit and simplify it.
Updated On: Oct 1, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Read the Circuit:
The figure has two parallel branches between the battery and the lamp.
Upper branch: \(S_1\) in series with a parallel pair \(S_2\) and \(S_3\). Expression: \(S_1(S_2+S_3)\).
Lower branch: \(S_3'\), \(S_2'\) and \(S_1\) in series. Expression: \(S_1S_2'S_3'\).
Step 2: Combine the Branches:
Parallel branches mean OR. So the circuit is
\[ S_1(S_2+S_3)+S_1S_2'S_3' = S_1\left[S_2+S_3+S_2'S_3'\right] \]
Step 3: Simplify:
By De Morgan, \(S_2'S_3'=(S_2+S_3)'\). So \(S_2+S_3+(S_2+S_3)'=1\) (a variable plus its complement is 1).
\[ S_1\cdot1=S_1 \]
Step 4: Choose the Option:
The simplified circuit is just a single switch \(S_1\) in series with the lamp. Option (A) shows exactly that. Option (B) shows only \(S_2\), (C) shows \(S_1\) and \(S_2\) in parallel, and (D) shows them in series, none of which equals \(S_1\) alone.
Final Answer:
The circuit reduces to the single switch \(S_1\), option (A). \[ \boxed{\text{(A) Single switch } S_1} \]
The figure has two parallel branches between the battery and the lamp.
Upper branch: \(S_1\) in series with a parallel pair \(S_2\) and \(S_3\). Expression: \(S_1(S_2+S_3)\).
Lower branch: \(S_3'\), \(S_2'\) and \(S_1\) in series. Expression: \(S_1S_2'S_3'\).
Step 2: Combine the Branches:
Parallel branches mean OR. So the circuit is
\[ S_1(S_2+S_3)+S_1S_2'S_3' = S_1\left[S_2+S_3+S_2'S_3'\right] \]
Step 3: Simplify:
By De Morgan, \(S_2'S_3'=(S_2+S_3)'\). So \(S_2+S_3+(S_2+S_3)'=1\) (a variable plus its complement is 1).
\[ S_1\cdot1=S_1 \]
Step 4: Choose the Option:
The simplified circuit is just a single switch \(S_1\) in series with the lamp. Option (A) shows exactly that. Option (B) shows only \(S_2\), (C) shows \(S_1\) and \(S_2\) in parallel, and (D) shows them in series, none of which equals \(S_1\) alone.
Final Answer:
The circuit reduces to the single switch \(S_1\), option (A). \[ \boxed{\text{(A) Single switch } S_1} \]
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