Asymmetry of Unions and Intersections

[From my discussion with Prof. Green]

We know the usual result that most (if not all) theoretic identity using unions, intersections and complements has a corresponding dual with unions and intersections exchanged and complements unchanged. This indicates some kind of symmetry between the union and intersection operations. For example, the pairs of de'morgan's laws, distribution of union over intersection and vice versa.

But union and intersection are not completely symmetric in the following sense.

A.B <= A and A+B >= A

So union typically creates a bigger set and intersection typically creates a smaller set.

Now there is something called the smallest set - the empty set. You cannot get any smaller than that.

But Cantor's famous result implies that there is no biggest set; given any set, one can construct another set (its he power set) which is strictly bigger.

Thus we have some sort of asymmetry for unions and intersections.

So the question is, does this translate into an identity that is not valid when the unions and intersections are exchanged?