Electrical – Can we derive all boolean functions using a 2×1 Multiplexer

boolean-algebradigital-logicmultiplexer

It is possible to implement all boolean functions using a 4×1, 8×1 or a 16×1 Multiplexer.

But is it possible to implement all boolean functions with a 2×1 Multiplexer? I think it is not, because how would it be possible to implement AND, OR, XOR, etc. I think only NOR can be implemented using a 2×1 Multiplexer.

Am I missing any point here?

Best Answer

Starting from a standard 2x1 mux:

schematic

simulate this circuit – Schematic created using CircuitLab

You can write its output \$O\$ as a function of \$A, B, C\$: $$ O = A\cdot\overline{C} + B\cdot C $$

Starting from here, you have a lot of possibilities. $$ A = 1, B = 0 \implies O = \overline{C}\\ B = 1 \implies O = A\cdot\overline{C} + 1\cdot C = A + C\\ A = 0 \implies O = B\cdot C $$

So you already have not, or, and. I think you can work the rest from here.

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