# 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 $\overline{\overline{A\overline{AB}}\,\overline{\overline{AB}B}} ={A\overline{AB}+\overline{AB}B} =A(\overline{A} + \overline{B})+(\overline{A} + \overline{B})B =A\overline B+\overline{A}B$? – John L. Apr 16 '19 at 12:12
• Yes, thankyou @Apass.Jack – Shane Apr 16 '19 at 13:20
• $\overline{\overline{A + \overline{A+B}}+\overline{\overline{A+B}+B}} =(A+\overline{A+B})(\overline{A+B}+B) =(A+\overline{A}\,\overline{B})(\overline{A}\,\overline{B}+B) =(AB+\overline{A}\,\overline{B})$. Then, $\overline{(AB+\overline{A}\,\overline{B}) + (AB+\overline{A}\,\overline{B})} =\overline{(AB+\overline{A}\,\overline{B})} =\overline{AB}\,\overline{\overline{A}\,\overline{B}} =\overline{AB}(A+B) =(\overline{A} + \overline{B})(A+B) =\overline{A}B + \overline{B}A =A\overline{B} + B\overline{A}$ – John L. Apr 23 '19 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.