Create an XOR gate using only NAND gates.
Solution
By definition:
a XOR b = (a OR b) AND NOT(a AND B)
The key is to apply De Morgan's rule repeatedly:
NOT(x OR y) = (NOT x) AND (NOT y)
NOT(x AND y) = (NOT x) OR (NOT y)
and the fact that:
NOT(x AND y) = x NAND y
NOT x = x NAND x
until XOR can be expressed entirely as NOT and NAND operations:
a XOR b
= (a OR b) AND (a NAND b)
= (NOT(NOT(a OR b))) AND (a NAND b)
= (NOT((NOT a) AND (NOT b))) AND (a NAND b)
= ((NOT a) NAND (NOT b)) AND (a NAND b)
= NOT(((NOT a) NAND (NOT b)) NAND (a NAND b))
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