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 logical expression.
We are given the logical expression:
\[ \left[ (q \land ( \sim q \lor r)) \land (\sim p \lor (p \land r)) \right]. \]
This is a combination of AND (\( \land \)) and OR (\( \lor \)) operations. We need to simplify this expression to identify the correct circuit diagram.

Step 2: Simplifying the logical expression.
First, simplify the expression \( (q \land (\sim q \lor r)) \). Using the distributive property:
\[ q \land (\sim q \lor r) = (q \land \sim q) \lor (q \land r). \]
Since \( q \land \sim q = \text{false} \), the expression simplifies to: \[ q \land r. \]
Next, simplify the expression \( (\sim p \lor (p \land r)) \). Using the distributive property:
\[ \sim p \lor (p \land r) = (\sim p \lor p) \land (\sim p \lor r). \]
Since \( \sim p \lor p = \text{true} \), the expression simplifies to:
\[ \sim p \lor r. \]

Step 3: Final simplification.
Now the expression becomes:
\[ (q \land r) \land (\sim p \lor r). \]
Using the distributive property again: \[ (q \land r) \land (\sim p \lor r) = (q \land r \land \sim p) \lor (q \land r \land r). \]
Since \( r \land r = r \), the expression simplifies to:
\[ (q \land r \land \sim p) \lor (q \land r). \]
This is the simplified logical expression.

Step 4: Identifying the correct circuit diagram.
Based on the simplified expression, the correct circuit diagram corresponds to option (A), which matches the logic gates required to implement the simplified expression.
Final Answer:
The correct simplified circuit diagram is:
\[ \boxed{(A)}. \]