Q.Realize XNOR gate using only NOR gates.

The XNOR function $Y = \overline{A \oplus B} = AB + \overline{A}\,\overline{B}$ can be built from four NOR gates by first forming $\overline{A+B}$ and using it to generate the two required terms. …

[!TLDR]

Four NOR gates realise XNOR: form $\overline{A+B}$, use it to make $\overline{A}B$ and $A\overline{B}$, then NOR those two to get $\overline{A \oplus B}$.

Since NOR is a universal gate, the exclusive-NOR function can be produced from NOR gates only. The target is $Y = \overline{A \oplus B} = AB + \overline{A}\,\overline{B}$. The trick is that a NOR gate fed with an input and the common term $\overline{A+B}$ produces exactly the individual product terms needed.

Take the four NOR gates in sequence:

• $G_1 = \overline{A + B}$
• $G_2 = \overline{A + G_1} = \overline{A}\,\overline{G_1} = \overline{A}\,(A + B) = \overline{A}B$
• $G_3 = \overline{B + G_1} = \overline{B}\,(A + B) = A\overline{B}$
• $G_4 = \overline{G_2 + G_3} = \overline{\,\overline{A}B + A\overline{B}\,} = \overline{A \oplus B}$ …