What does "dual" mean exactly in mathematics? I'm not a math expert but I know a little bit of calculus and theorems. I've heard things like "this result is "dual"", or this "theorem is "dual"". Often people say "dual comes for free". Like you swap variables or something and you get another result.
I've never understood this. Can anyone explain what "dual" means precisely? Examples would be nice as well.
Thanks!
 A: Broadly, two mathematical set-ups are dual when they can be transformed into each other by a simple exchange of symbols and terminology. For example, a general topological space may be defined in terms of its open sets, or dually in terms of its closed sets: corresponding to infinite unions and finite intersections of open sets, in the first formulation, are respectively finite unions and infinite intersections in the second one. Another example is plane projective geometry: a theorem about lines through points, and points where lines intersect, can be transformed into a distinct dual theorem, respectively about points where lines intersect and lines through points. The dual of a finite-dimensional vector space is the space of linear functionals on it; and the dual of the latter space is isomorphic to the original space via a canonical mapping.
Unfortunately, the word dual is not confined to such simple cases where, if B is the dual of A, then A is the dual of B. For example, in the case of infinite-dimensional vector spaces, the dual of a space (defined as above) can be much "bigger" than the original, and its dual in turn is bigger still. However, when the word is used, there is usually a clear link to the simpler idea of duality.
A: A duality is a pair of related concepts that display a one-to-one translation symmetry, usually (not always) as the result of some form of involution operator.
In classical logic, the operators, $\vee$ and $\wedge$ form a dual, and negation is their involution operator.   This is expressed through deMorgan's Laws:$$\neg(A\vee B) = \neg A\wedge \neg B\\\neg(A\wedge B)=\neg A\vee \neg B$$
Similarly in set algebra, the $\cup$ and $\cap$ operators form a dual, with complementation being the involution. $$(A\cup B)^\complement = A^\complement\cap B^\complement\\ (A\cap B)^\complement=A^\complement\cup B^\complement$$
In predicate logic, the quantifiers, $\forall$ and $\exists$ are a dual, again with negation being the transformation.  $$\neg \forall x\;P(x) \iff \exists x\;\neg P(x)\\\neg \exists x\;P(x) \iff \forall x\;\neg P(x)$$
And such.
