GATE 2022 IN Q.21: solution and answer

GATE 2022 · INQ.21 · 1 mark · MCQ

The logic block shown has an output $F$ given by ________

GATE 2022 IN question 21: Two inputs, B (upper) and A (lower), enter from the left as horizontal lines. — Figure from the GATE 2022 IN paper. It is part of the question.

(A)
$A + B$
(B)
$A . \overline{B}$
(C)
$A + \overline{B}$
(D)
$\overline{B}$

Answer and solution

ANSWER

$F = \overline{B}$ , which is option (D).

Official answer: DSource: Official answer key, GATE 2022 IN (IIT Kharagpur)

Read the rails from top to bottom. The two inverters give $\overline{B}$ on the top rail and $\overline{A}$ on the third rail, and $B$ and $A$ run through unchanged on the second and fourth rails. Here $\overline{X}$ means the complement of $X$ . A dot marks a connection. Where a vertical wire crosses a rail with no dot, the two are not connected.
Left NOR gate. Its inputs are tapped (dots) from the $\overline{A}$ rail and the $\overline{B}$ rail. By De Morgan's theorem: $X = \overline{\overline{A} + \overline{B}} = \overline{\overline{A}} \cdot \overline{\overline{B}} = A B$
Right NOR gate. Its inputs are tapped (dots) from the $A$ rail and the $\overline{B}$ rail. By De Morgan's theorem: $Y = \overline{A + \overline{B}} = \overline{A} \cdot \overline{\overline{B}} = \overline{A} B$
Output NOR gate, fed by $X$ and $Y$ : $F = \overline{X + Y} = \overline{A B + \overline{A} B}$
Factor out $B$ and use the complement law $A + \overline{A} = 1$ : $A B + \overline{A} B = B (A + \overline{A}) = B$
So $F = \overline{B}$ , which is option (D). Check with $A = 0, B = 0$ : $X = 0$ and $Y = 0$ , so $F = \overline{0} = 1 = \overline{B}$ . Check with $A = 1, B = 1$ : $X = 1$ , so $F = 0 = \overline{B}$ . Both agree.

KEY IDEA

A NOR gate fed with complemented inputs acts as an AND gate: $\overline{\overline{P} + \overline{Q}} = P Q$ . So the two first-level NOR gates produce the product terms $A B$ and $\overline{A} B$ . The output NOR gives the complement of their sum. Since $A B + \overline{A} B = B$ , the output does not depend on $A$ and $F = \overline{B}$ .

WATCH OUT FOR

Treating a crossing wire as a connection. The vertical wires from the $\overline{B}$ rail cross the $B$ , $\overline{A}$ and $A$ rails with no dot, so they connect only to $\overline{B}$ . If you read the right gate's inputs as $A$ and $B$ , you get $F = A \oplus B$ , which is not one of the options. The other slip is dropping the output bubble on the last gate. That gives $F = B$ instead of $\overline{B}$ .

From the GATE 2022 IN paper. Every question of this paper · Download the full paper · All 22 Boolean logic questions, with notes

GATE 2022 IN Q.21: solution and answer

Answer and solution

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