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 $\overline{\overline{A\overline{AB}}\,\overline{\overline{AB}B}} ={A\overline{AB}+\overline{AB}B} =A(\overline{A} + \overline{B})+(\overline{A} + \overline{B})B =A\overline B+\overline{A}B$? – Apass.Jack Apr 16 at 12:12
• Yes, thankyou @Apass.Jack – Shane Apr 16 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, $\overline{(AB+\overline{A}\,\overline{B}) + (AB+\overline{A}\,\overline{B})} =\overline{(AB+\overline{A}\,\overline{B})} =\overline{AB}\,\overline{\overline{A}\,\overline{B}} =\overline{AB}(A+B) =(\overline{A} + \overline{B})(A+B) =\overline{A}B + \overline{B}A =A\overline{B} + B\overline{A}$ – Apass.Jack Apr 23 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.