Find the Boolean expression for the output Y if inputs are A and B.

Question

Examoogle · Accepted Answer

The correct answer is Option A: $Y = \overline{A \cdot B} + \overline{A + B}$. To determine the output $Y$, we derive the Boolean expression gate-by-gate. By applying Boolean laws like De Morgan's theorem and simplification rules, the final circuit output simplifies to $Y = B$.

Step-by-Step Solution & Concepts:
To determine the output $Y$, we derive the Boolean expression gate-by-gate. By applying Boolean laws like De Morgan's theorem and simplification rules, the final circuit output simplifies to $Y = B$.

Working Principle: The circuit consists of a NAND gate, two NOR gates, and an OR gate. Let the top NAND gate have output $\overline{AB}$. Let the bottom NOR gate have input $B$ and $A$, yielding $\overline{A+B}$. The next level processes these signals through further NOR operations before finally reaching the OR gate. Simplifying the Boolean algebra step-by-step: $Y = (\overline{\overline{AB} + \overline{A+B}}) + (\overline{\overline{A+B} + A})$. Further reduction leads to the result $Y = B$.

Key Formulas: $Y = A \cdot B$ — NAND gate operation; $Y = \overline{A+B}$ — NOR gate operation; $Y = A + B$ — OR gate operation

Key Points: De Morgan's theorem states $\overline{A+B} = \overline{A} \cdot \overline{B}$ and $\overline{A \cdot B} = \overline{A} + \overline{B}$.; Boolean absorption laws and involution laws simplify complex logical expressions.; Logic circuits can often be reduced to simpler equivalent expressions using algebraic manipulation.

Advantages: Reduces hardware complexity by identifying redundant logic gates.

Disadvantages: Complex analysis is required to verify the simplification of multi-gate circuits.

Applications: Digital logic design optimization; Microprocessor instruction decoding

Additional Info: The output of the circuit is independent of input A, making input A a 'don't care' condition for the final output Y.. Options B, C, and D are incorrect because they do not represent the final simplified logical equivalence of the given circuit diagram.

Answer: A is correct — The Boolean expression simplifies to $Y = B$ based on the logic circuit connectivity.

Exam Tip: Always verify truth tables for logic circuits if Boolean algebra simplification becomes too complex; mapping inputs A and B as (0,0), (0,1), (1,0), (1,1) will quickly reveal the dependency.

Examoogle · Answer

Option B: $Y = \overline{\overline{A \cdot B} + \overline{A + B}} + \overline{\overline{A + B} + A}$

Examoogle · Answer

Option C: $Y = \overline{\overline{A \cdot B} + \overline{A + B}} + \overline{\overline{A + B} + \overline{A + B}}$

Examoogle · Answer

Option D: $Y = (\overline{A \cdot B}) + (\overline{A + B})$

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