First, some definitions and set operations to take care of:

1. Nothing = Not anything = N

2. { } = The empty set = { }

3. A n B = The intersection of sets A and B

4. A u B = The union of sets A and B

5. A' = The complement of set A

6. H is a subset of A iff all the elements of set H are in set A

Suppose now that there's a universe U with only 4 things viz. v, w, x, and y. Suppose also that v and w are form the set A and x and y form the set B as below,

7. A = {v, w}

8. B = {x, y}

9. T = The universal set = A u B = {v, w, x, y}

10. In U, Nothing, as the definition asserts, is not v and not w and not x and not y. Basically, in terms of sets, Nothing is not A and not B.

Now,

not A = A' = {x, y}

not B = B' = {v, w}

Nothing = not A and not B = A' n B' = {x, y} n {v, w} = { } = not v and not w and not x and not y.

11. A' n B' = { }

However,

12. A' n B' yields the set whose elements are those that are common to both A' and B'. We now know that Nothing ( N ) is common to both A' and B'. In other words,

13. A' = {x, y, N}

and

14. B' = {v, w, N} and since N is common to both A' and B',

15. A' n B' ={N}

16. { } = {N} The empty set contains Nothing ( N ) [From 11 and 13]

17. (A u B)' = [{x, y, N} u {v, w, N}] ' = {v, w, x, y, N}'

and

18. (A u B)' = A' n B' (DeMorgan's law)

Therefore,

19. {v, w, x, y, N}' = {N} (from 15, 17, and 18)

Statement 19 is a paradox because the complement of a set can't contain an element that's in that set but in this case the element N = Nothing is both in the set (A u B) and also in (A u B)'. The Nothing-Empty Set Paradox.

P.S. To those who object to line 16. { } = {N}, remember that the empty set, { }, is a subset of every set and that means it must have an element that's in every other set. The only element that fits the bill is Nothing = N