Consider the following logic circuit diagram. Which is/are the CORRECT…
2025
Consider the following logic circuit diagram.
Which is/are the CORRECT option(s) for the output function 𝐹?
- A.
\(\overline{X Y}\) - B.
\(\overline{X}+\overline{Y}+X \overline{Y}\) - C.
\(\overline{XY}+\overline{X}+X \overline{Y}\) - D.
\(X+\overline{Y}\)
Attempted by 86 students.
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Correct answer: A, B, C
Key result: the output function F equals ¬(X·Y).
Step-by-step derivation:
Top gate: a NAND receiving X and Y, so its output is ¬(X·Y).
Middle inverter: inverts X, producing ¬X. This signal is one input to the final OR.
Bottom path: the inverter produces ¬X and the bottom AND receives X and ¬X, so that AND output is X·¬X = 0 (always false).
Final OR: combine the three inputs: ¬(X·Y) + ¬X + 0 = ¬(X·Y) + ¬X.
Use ¬(X·Y) = ¬X + ¬Y to see that ¬(X·Y) already includes ¬X, so the OR simplifies to ¬(X·Y).
Equivalence checks for the given algebraic expressions:
Expression ¬X + ¬Y + X·¬Y simplifies to ¬X + ¬Y (because ¬Y absorbs X·¬Y), which equals ¬(X·Y).
Expression ¬(X·Y) + ¬X + X·¬Y simplifies to ¬(X·Y) (because ¬(X·Y) = ¬X + ¬Y and that already covers the other terms).
Expression X + ¬Y is not equivalent; a counterexample is X=1, Y=1 where X+¬Y = 1 but ¬(X·Y) = 0.
Conclusion: the circuit implements F = ¬(X·Y). The two algebraic forms ¬X + ¬Y + X·¬Y and ¬(X·Y) + ¬X + X·¬Y are algebraically equivalent to ¬(X·Y), so they represent the same function; X + ¬Y does not.
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