# how do we demonstrate using Boolean algebra that these NAND and NOR gate combinations are XOR gates

How do we demonstrate using Boolean algebra that these NAND and NOR gate combinations are XOR gates?

• By boolean algebra, do you mean something like $\overline{\overline{A\overline{AB}}\,\overline{\overline{AB}B}} ={A\overline{AB}+\overline{AB}B} =A(\overline{A} + \overline{B})+(\overline{A} + \overline{B})B =A\overline B+\overline{A}B$? Commented Apr 16, 2019 at 12:12
• $\overline{\overline{A + \overline{A+B}}+\overline{\overline{A+B}+B}} =(A+\overline{A+B})(\overline{A+B}+B) =(A+\overline{A}\,\overline{B})(\overline{A}\,\overline{B}+B) =(AB+\overline{A}\,\overline{B})$. Then, $\overline{(AB+\overline{A}\,\overline{B}) + (AB+\overline{A}\,\overline{B})} =\overline{(AB+\overline{A}\,\overline{B})} =\overline{AB}\,\overline{\overline{A}\,\overline{B}} =\overline{AB}(A+B) =(\overline{A} + \overline{B})(A+B) =\overline{A}B + \overline{B}A =A\overline{B} + B\overline{A}$ Commented Apr 23, 2019 at 9:44

With two input wires, there are $$2^2=4$$ possible inputs to your circuit: $$00, 01, 10, 11$$. You can test each one and write down its output. Then compare that to the XOR gate's outputs.