Why is the fifth NOR (inputs tied) used?

To invert the XNOR output into XOR

Asked 2/22
2082 Shrawan · 3 marks

2-input XOR using NOR gates only

Short answer

An XOR gate can be implemented with five NOR gates. First, four NOR gates produce XNOR: G1 = (A+B)′, G2 = (A+G1)′, G3 = (B+G1)′, G4 = (G2+G3)′ = A⊙B. Then a fifth NOR with both inputs tied to G4 inverts it to yield Y = (G4+G4)′ = A⊕B.

Realize 2 inputs XOR gates using NOR gates only.

$Y = A \oplus B$ . Build XNOR with 4 NOR gates and invert it with a fifth NOR (inputs tied). Minimum: 5 NOR gates.

\begin{aligned} G_1 &= \overline{A + B} \\ G_2 &= \overline{A + G_1} = \bar A B \\ G_3 &= \overline{B + G_1} = A \bar B \\ G_4 &= \overline{G_2 + G_3} = \overline{A \oplus B} \\ Y &= \overline{G_4 + G_4} = A \oplus B \end{aligned}

 A --+-----------[NOR G2]--+
     |              |      |
     +--[NOR G1]----+      +--[NOR G4]--[NOR G5]-- Y
     |              |      |          (inputs tied)
 B --+-----------[NOR G3]--+

A	B	$G_1$	$G_2$	$G_3$	$G_4$	Y
0	0	1	0	0	1	0
0	1	0	1	0	0	1
1	0	0	0	1	0	1
1	1	0	0	0	1	0

✓ Answer checked by Claude Opus 5.5 (high effort)

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