Question

The correct simplified circuit diagram for the logical statement \[ \left[ (q \land ( \sim q \lor r)) \land (\sim p \lor (p \land r)) \right] \] is

• A. A
• B. B
• C. C
• D. D

Hint:

To simplify logical expressions, use properties like distributive, associative, and commutative laws, and apply them step by step to break down complex expressions into simpler forms.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to simplify the complex logical expression using distributive, absorption, and other laws of logic.
Step 2: Detailed Explanation:
1. Simplify \( q \land (\sim q \lor r) \):
\( (q \land \sim q) \lor (q \land r) = \text{F} \lor (q \land r) = q \land r \).
2. Simplify \( \sim p \lor (p \land r) \):
\( (\sim p \lor p) \land (\sim p \lor r) = \text{T} \land (\sim p \lor r) = \sim p \lor r \).
3. Combine the first two parts:
\( (q \land r) \land (\sim p \lor r) = q \land (r \land (\sim p \lor r)) \).
Since \( r \land (\sim p \lor r) = r \) (Absorption law), we get \( q \land r \).
4. Now the whole expression:
\( (q \land r) \lor (p \land r) \).
By taking \( r \) common (Distributive law):
\( (q \lor p) \land r \).
The simplified statement is \( (p \lor q) \land r \).
The corresponding circuit has \( p \) and \( q \) in parallel, and both together in series with \( r \).
Step 3: Final Answer:
The circuit consists of \( p \) and \( q \) in parallel, connected in series with \( r \).