Signed sets may be represented mathematically as an ordered pair of disjoint sets, one set for their positive elements and another for their negative elements.
[1] Alternatively, they may be represented as a Boolean function, a function whose domain is the underlying unsigned set (possibly specified explicitly as a separate part of the representation) and whose range is a two-element set representing the signs.
[2] Signed sets are fundamental to the definition of oriented matroids.
If the hypercube consists of all points in Euclidean space of a given dimension whose Cartesian coordinates are in the interval
This subset of points forms a face, whose codimension is the cardinality of the signed subset.
[4] The number of signed subsets of a given finite set of
, a power of three, because there are three choices for each element: it may be absent from the subset, present with positive sign, or present with negative sign.
[5] For the same reason, the number of signed subsets of cardinality
is and summing these gives an instance of the binomial theorem, An analogue of the Erdős–Ko–Rado theorem on intersecting families of sets holds also for signed sets.
According to this theorem, for any a collection of signed subsets of an
and all pairs having a non-empty intersection, the number of signed subsets in the collection is at most For instance, an intersecting family of this size can be obtained by choosing the sign of a single fixed element, and taking the family to be all signed subsets of cardinality
this theorem follows immediately from the unsigned Erdős–Ko–Rado theorem, as the unsigned versions of the subsets form an intersecting family and each unsigned set can correspond to at most