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Not Another NAND!

Age 14 to 18 Challenge Level:

 

The NAND gate is said to be universal. This means that the effect of any other logic gate can be replicated using just NAND gates.

Prove that this is indeed the case: replicate AND, OR, XOR, XNOR, NOR, NOT gates using just NAND gates.

Are any of the other types of gate universal? Prove it.