  A physical system is synergistic if it cannot be reduced to its constituents.
Intuitively this is paraphrased into the common statement that 'the whole is
greater than the sum of its parts'. In this manner, many basic elements in
combination may give rise to some unexpected collective behavior. A
paradigmatic example of such phenomenon is information. Several sources, which
are already known individually, may provide some new knowledge when joined
together. Here we take the trivial case of discrete random variables and
explore whether and how it is possible get more information out of lesser
parts. Our approach is inspired by set theory as the fundamental description of
part-whole relations. If taken unaltered, synergistic behavior is forbidden by
the set theoretical axioms. Indeed, the union of sets cannot contain extra
elements not found in any particular one of them. However, random variables are
not a perfect analogy of sets. We formalise the distinction, finding a single
broken axiom - union/intersection distributivity. Nevertheless, it remains
possible to describe information using Venn-type diagrams. We directly connect
the existence of synergy to the failure of distributivity for random variables.
When compared to the partial information decomposition framework (PID), our
technique fully reproduces previous results while resolving the
self-contradictions that plagued the field and providing additional constraints
on the solutions. This opens the way towards quantifying emergence in large
systems.
