Question:
Simplest form of the following switching circuit is
Simplest form of the following switching circuit is
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Write the Boolean expression of the circuit and simplify.
Updated On: Oct 1, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Reading the circuit:
The figure shows three parallel branches between the battery and the lamp. Branch 1 has \(S_1\) and \(S_2\) in series, branch 2 has \(S_1\) and \(S_2'\) in series, and branch 3 has \(S_1'\) and \(S_2\) in series.
Step 2: Boolean expression:
Series means AND and parallel means OR:
\[ (S_1 \wedge S_2) \vee (S_1 \wedge S_2') \vee (S_1' \wedge S_2) \]
Step 3: Simplify:
Combine the first two terms: \(S_1 \wedge (S_2 \vee S_2') = S_1 \wedge T = S_1\).
The expression becomes \(S_1 \vee (S_1' \wedge S_2)\).
By the absorption identity \(p \vee (\sim p \wedge q) = p \vee q\), this is \(S_1 \vee S_2\).
Step 4: Final circuit:
\(S_1 \vee S_2\) is a circuit with \(S_1\) and \(S_2\) in parallel. This matches option (B), where two switches are drawn on parallel branches. Option (A) is a series circuit, (C) is \(S_1 \wedge S_2'\), and (D) is \(S_1 \vee S_2'\).
Final Answer:
The circuit reduces to S1 or S2, two switches in parallel. \[ \boxed{\text{(B) }\text{S}_1\ \text{and}\ \text{S}_2\ \text{in parallel}} \]
The figure shows three parallel branches between the battery and the lamp. Branch 1 has \(S_1\) and \(S_2\) in series, branch 2 has \(S_1\) and \(S_2'\) in series, and branch 3 has \(S_1'\) and \(S_2\) in series.
Step 2: Boolean expression:
Series means AND and parallel means OR:
\[ (S_1 \wedge S_2) \vee (S_1 \wedge S_2') \vee (S_1' \wedge S_2) \]
Step 3: Simplify:
Combine the first two terms: \(S_1 \wedge (S_2 \vee S_2') = S_1 \wedge T = S_1\).
The expression becomes \(S_1 \vee (S_1' \wedge S_2)\).
By the absorption identity \(p \vee (\sim p \wedge q) = p \vee q\), this is \(S_1 \vee S_2\).
Step 4: Final circuit:
\(S_1 \vee S_2\) is a circuit with \(S_1\) and \(S_2\) in parallel. This matches option (B), where two switches are drawn on parallel branches. Option (A) is a series circuit, (C) is \(S_1 \wedge S_2'\), and (D) is \(S_1 \vee S_2'\).
Final Answer:
The circuit reduces to S1 or S2, two switches in parallel. \[ \boxed{\text{(B) }\text{S}_1\ \text{and}\ \text{S}_2\ \text{in parallel}} \]
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