De Morgan's laws; XNOR using NOR gates

De Morgan's laws

First theorem: the complement of a sum (OR) of variables equals the product (AND) of their complements.

$$\overline{A + B} = \bar{A} \cdot \bar{B}$$

Second theorem: the complement of a product (AND) of variables equals the sum (OR) of their complements.

$$\overline{A \cdot B} = \bar{A} + \bar{B}$$

In words: "break the bar and change the sign".

Example: $\overline{\bar A + B} = A \bar B$ and $\overline{A \bar C} = \bar A + C$.

2-input XNOR using only NOR gates

$Y = A \odot B = AB + \bar A \bar B = \overline{A\bar B + \bar A B}$ needs four 2-input NOR gates (the dual of the 4-NAND XOR).

$$\begin{aligned} G_1 &= \overline{A + B} \\ G_2 &= \overline{A + G_1} = \bar A (A + B) = \bar A B \\ G_3 &= \overline{B + G_1} = \bar B (A + B) = A \bar B \\ Y &= \overline{G_2 + G_3} = \overline{\bar A B + A \bar B} = A \odot B \end{aligned}$$

 A --+---------------[NOR G2]--+
     |                 |       |
     +--[NOR G1]---G1--+       +--[NOR G4]-- Y
     |                 |       |
 B --+---------------[NOR G3]--+

Y = 1 only when A = B, which is the XNOR function.