In mathematics, a signed set is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of elements together with an assignment of a

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

(positive or negative) to each element of the set.

Representation

Signed sets may be represented mathematically as an

ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...

disjoint sets In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...

, one set for their positive elements and another for their negative elements. Alternatively, they may be represented as a

Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth functi ...

, 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. Signed sets may also be called

\Z_2

- graded sets.

Application

Signed sets are fundamental to the definition of oriented matroids. They may also be used to define the faces of a

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

. If the hypercube consists of all points in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

of a given dimension whose

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

are in the interval

1,+1 Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 399 at the 2020 census. The village is located on the northeast shore of Portage Lake and is surrounded by Onekama Township. The town's name is deri ...

/math>, then a signed subset of the coordinate axes can be used to specify the points whose coordinates within the subset are

-1

+1

(according to the sign in the signed subset) and whose other coordinates may be anywhere in the interval

/math>. This subset of points forms a face, whose

codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...

is the

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

of the signed subset.

Combinatorics

Enumeration

The number of signed subsets of a given

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...

n

elements is

3^n

, a

power of three In mathematics, a power of three is a number of the form where is an integer, that is, the result of exponentiation with number 3, three as the Base (exponentiation), base and integer as the exponent. The first seven non-negative powers ...

, because there are three choices for each element: it may be absent from the subset, present with positive sign, or present with negative sign. For the same reason, the number of signed subsets of cardinality

r

is :

2^r\binom,

and summing these gives an instance of the

binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...

, :

\sum_r 2^r\binom=3^n.

Intersecting families

An analogue of the Erdős–Ko–Rado theorem on intersecting families of sets holds also for signed sets. The intersection of two signed sets is defined to be the signed set of elements that belong to both and have the same sign in both. According to this theorem, for any a collection of signed subsets of an

n

-element set, all having cardinality

r

and all pairs having a non-empty intersection, the number of signed subsets in the collection is at most :

2^\binom.

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

r

that contain this element with this sign. For

r\le n/2

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

2^

signed sets. However, for larger values of

r

a different proof is needed.

References

{{reflist, refs= {{citation , last = Brini , first = A. , at = Art. B55g , date = July 2005 , journal = Séminaire Lotharingien de Combinatoire , mr = 2373407 , title = Combinatorics, superalgebras, invariant theory and representation theory , url = https://eudml.org/doc/227556 , volume = 55; see in particular Section 3.4, p. 15 {{citation , last1 = Bollobás , first1 = B. , author1-link = Béla Bollobás , last2 = Leader , first2 = I. , author2-link = Imre Leader , doi = 10.1016/S0898-1221(97)00215-0 , issue = 11 , journal = Computers and Mathematics with Applications , mr = 1486880 , pages = 9–13 , title = An Erdős–Ko–Rado theorem for signed sets , volume = 34 , year = 1997, doi-access = free This formula for the number of signed subsets and the number of faces of a hypercube generalizes to the number of faces of a

Hanner polytope In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Swedish mathematician Olof Hanner, who introduced them in 1956.. Construction The Hann ...

; see {{citation , last = Kalai , first = Gil , authorlink = Gil Kalai , doi = 10.1007/BF01788696 , issue = 1 , journal = Graphs and Combinatorics , mr = 1554357 , pages = 389–391 , title = The number of faces of centrally-symmetric polytopes , volume = 5 , year = 1989 {{citation , last = Las Vergnas , first = Michel , authorlink = Michel Las Vergnas , doi = 10.1016/0095-8956(80)90082-9 , issue = 2 , journal =

Journal of Combinatorial Theory The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applicati ...

, mr = 586435 , pages = 231–243 , series = Series B , title = Convexity in oriented matroids , volume = 29 , year = 1980, doi-access = free {{citation , last1 = Metropolis , first1 = N. , author1-link = Nicholas Metropolis , last2 = Rota , first2 = Gian-Carlo , author2-link = Gian-Carlo Rota , doi = 10.1090/S0002-9904-1978-14477-2 , issue = 2 , journal =

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

, mr = 462997 , pages = 284–286 , title = On the lattice of faces of the

n

-cube , volume = 84 , year = 1978, doi-access = free Set theory