set
A finite set of numbers, text, or nested sets. Duplicate elements are removed. Numeric sets are sorted; mixed sets retain their first-occurrence order.
xs(array) --
Sets
union
A union B.
A(set) --B(set) --
difference
A minus B: the elements of A not in B.
A(set) --B(set) --
symmetricDifference
Elements in exactly one of A and B.
A(set) --B(set) --
setWhere
Select members of a finite numeric universal set for which an exact rule is true.
universe(set) -- the finite candidate setrule(string) -- a numeric conditionvariable(string) -- the variable bound to each candidate
interval
A real interval with exact finite endpoints; -inf and inf give unbounded ends. Each finite end is open or closed.
lower(number or -inf) --upper(number or inf) --lowerBoundary(string) -- open or closedupperBoundary(string) -- open or closed
complement
The complement of A within the universal set U.
U(set) --A(set) --
isSubset
True if every element of A is in B.
A(set) --B(set) --
isSuperset
True if A contains every element of B.
A(set) --B(set) --
isDisjoint
True if A and B have no element in common.
A(set) --B(set) --
setEqual
True if A and B have the same elements.
A(set) --B(set) --
contains
True if x is an element of A.
A(set) --x(number, text, or set) --
cardinality
The number of elements, |A|.
A(set) --
powerSet
The set of all subsets of A, shown as a set of sets with the subsets in order of size. cardinality and contains work on it. At most 20 elements.
A(set) --
cartesianProduct
The Cartesian product A x B as a set of ordered pairs (a, b), a in A and b in B; with three sets, the set of triplets. The set functions (cardinality, contains, isSubset, union) work on it.
A(set) --B(set) --C(set) -- optional: a third set, for triplets
pair
The ordered pair (a, b). Order counts: pair(1, 2) is not pair(2, 1). A set of pairs is a relation.
a(number, text or set) --b(number, text or set) --
triplet
The ordered triplet (a, b, c).
a(number, text or set) --b(number, text or set) --c(number, text or set) --
relationWhere
The relation from A to B given by a rule: the pairs (x, y) of A x B for which the rule holds.
A(set) -- of numbersB(set) -- of numbersrule(string) -- a condition on the two namesx(string) -- the name the rule uses for the element of Ay(string) -- the name it uses for the element of B
domain
The domain of a relation: the set of first components of its pairs.
R(relation) -- a set of ordered pairs
range
The range of a relation: the set of second components of its pairs. (With numbers, range(a, b) is the list a, ..., b - 1.)
R(relation) -- a set of ordered pairs
imageOf
The set of images of x under a relation: every y with (x, y) in R. A function gives one image.
R(relation) -- a set of ordered pairsx(number, text or set) --
inverseRelation
The relation with every pair turned round: (y, x) for each (x, y).
R(relation) -- a set of ordered pairs
isRelation
Whether R is a relation from A to B: a subset of A x B.
R(relation) -- a set of ordered pairsA(set) --B(set) --
isFunction
Whether a relation is a function: no element has two images. With a set A, whether it is a function from A: every element of A has exactly one image.
R(relation) -- a set of ordered pairsA(set) -- optional: the set it must be defined on
isOneOne
Whether a relation is one-one: different elements never have the same image.
R(relation) -- a set of ordered pairs
isOnto
Whether a relation is onto B: every element of B is an image.
R(relation) -- a set of ordered pairsB(set) --
isReflexive
Whether (x, x) is in R for every x of A.
R(relation) -- a set of ordered pairsA(set) --
isSymmetric
Whether (y, x) is in R whenever (x, y) is.
R(relation) -- a set of ordered pairs
isTransitive
Whether (x, z) is in R whenever (x, y) and (y, z) are.
R(relation) -- a set of ordered pairs
isEquivalence
Whether R is an equivalence relation on A: reflexive, symmetric and transitive.
R(relation) -- a set of ordered pairsA(set) --
reflexiveClosure
A relation with (x, x) added for every element; the result is a set of pairs.
R(relation) -- a set of pairs, or a list such as [[1, 2], [2, 3]]
symmetricClosure
A relation with (b, a) added for every (a, b).
R(relation) -- a set of pairs, or a list such as [[1, 2], [2, 3]]
transitiveClosure
A relation with (a, c) added whenever (a, b) and (b, c) are in it.
R(relation) -- a set of pairs, or a list such as [[1, 2], [2, 3]]
equivalenceClasses
The classes of the equivalence relation generated by a list of pairs.
R(list of pairs) --