A system of four gates is set up as shown. The 'truth table' corresponding to this system is :

First Gate ($$G_1$$): This is a standard NOR gate taking inputs $$A$$ and $$B$$. Output of $$G_1 = \overline{A+B}$$

Second Gate ($$G_2$$): Takes input $$A$$ and the output of $$G_1$$. Output of $$G_2 = \overline{A + \overline{A+B}}$$

Applying De Morgan’s Law: $$\overline{A} \cdot (A+B) = \overline{A}A + \overline{A}B = 0 + \overline{A}B = \mathbf{\overline{A}B}$$

Third Gate ($$G_3$$): Takes input $$B$$ and the output of $$G_1$$. Output of $$G_3 = \overline{B + \overline{A+B}}$$

Applying De Morgan’s Law: $$\overline{B} \cdot (A+B) = \overline{B}A + \overline{B}B = \mathbf{A\overline{B}} + 0$$

Final Gate ($$G_4$$): Takes the outputs of $$G_2$$ and $$G_3$$ as its inputs. Output $$Y = \overline{\overline{A}B + A\overline{B}}$$

$$Y = \overline{A \oplus B} = A \odot B \text{ (XNOR)}$$

The XNOR gate is yields a 1 only when both inputs are the same and a 0 when they differ.