Question

show that the dual of the exclusive or is equal to its complement

Answer 1

First, rewrite p xor q as p'q + pq'. They are equivalent because their respective truth tables are equivalent.

Second, let's take the complement of p'q + pq' = (p'q + pq')', which by De Morgan's law (or by considering a Venn diagram) is (p'q)'(pq')'. Applying De Morgan's law again, we get (p + q')(p' + q).

Now, let's consider the dual of p'q + pq'. This is just replacing a union operator with an intersection operator and vice versa. So, that gives us (p' + q)(p + q') = (p + q')(p' + q). We see that the complement of p'q + pq' is the same as the dual of p'q + pq', thus xor's complement and dual are equivalent.