set6 is an object-oriented interface for constructing and manipulating mathematical sets using R6. set6 allows a variety of mathematical sets, including Sets, Tuple, Intervals and Fuzzy variants; there are also Conditional sets for creating sets out of complex logical instructions. A full set of tutorials can be found here. In this introductory vignette we briefly demonstrate how to construct a set, access its properties and traits, and some basic algebra of sets.
Before we get started note that by default set6
uses unicode for printing representations of sets. Whilst this is preferred for neater printing, some machines, operating systems, or R versions may not behave as expected when printing unicode and therefore we allow the option to turn this off.
useUnicode(FALSE)
This vignette does use the unicode representations.
Classes in set6
can be split into two groups: set objects, and wrappers. When we refer to sets (lower-case ‘s’) we refer to objects inheriting from the base class Set
. All sets in set6
inherit from Set
, meaning that, at a very minimum, they share the same methods and fields as Set
.
The simplest set of all is the empty set, we can see how even this has methods for printing and summarising, as well as a list of properties and traits.
empty = Set$new() empty #> ∅ summary(empty) #> Set #> ∅ #> Traits: #> Crisp #> Properties: #> Empty #> Cardinality = 0 - countably finite #> Closed empty$properties #> $empty #> [1] TRUE #> #> $singleton #> [1] FALSE #> #> $cardinality #> [1] 0 #> #> $countability #> [1] "countably finite" #> #> $closure #> [1] "closed" empty$traits #> $crisp #> [1] TRUE
Sets can take elements of any class, including other sets.
Set$new(1,2,3) #> {1, 2, 3} Set$new(letters[1:5]) #> {a, b,...,d, e} Set$new(1, 2i, "a", Set$new(1)) #> {1, 0+2i, a, {1}}
Each set class has its own unique mathematical properties, we will not cover these extensively here but summarise each with a short example.
# A Set cannot have duplicated elements, and ordering does not matter Set$new(1,2,2,3) == Set$new(3,2,1) #> [1] TRUE # A Tuple can have duplicated elements, and ordering does matter Tuple$new(1,2,2,3) != Tuple$new(1,2,3) #> [1] TRUE Tuple$new(1,3) != Tuple$new(3,1) #> [1] TRUE # An interval can be an interval of integers or numerics, but must be continuous Interval$new(1, 10, class = "integer") #> {1,...,10} Interval$new(1, 10) # numeric is default #> [1,10] # `type` is used to specify the interval upper and lower closure Interval$new(1, 10, type = "()") #> (1,10) # SpecialSets are useful for common 'special' mathematical sets # Use listSpecialSets() to see which are available. Reals$new() #> ℝ PosIntegers$new() #> ℤ+ # ConditionalSets are user-built sets from logical statements. # For example, the set of even numbers. ConditionalSet$new(function(x) x %% 2 == 0) #> {x%%2 == 0 : x ∈ V} # Finally FuzzySets and FuzzyTuples expand Sets and Tuples to allow for partial # membership of elements. These have two constructors, either by specifying the elements # and membership alternatively, or by passing these separately to the given arguments. FuzzySet$new(1, 0.1, 2, 0.2, "a", 0.3) == FuzzySet$new(elements = c(1,2,"a"), membership = c(0.1,0.2,0.3)) #> [1] TRUE
Every set has methods for comparing it to other sets, as well as for checking which elements are contained within in. Operators are overloaded where possible, and where not other infix operators are defined, these are:
$isSubset(x, proper = TRUE)
(<
) - Is x
a proper subset of self
$isSubset(x, proper = FALSE)
(<=
) - Is x
a (non-proper) subset of self
$isSubset(x, proper = TRUE)
(>
) - Is self
a proper subset of x
$isSubset(x, proper = FALSE)
(>=
) - Is self
a (non-proper) subset of x
$equals(x)
(==
) - Is x
(mathematically) equal to self
!($equals(x))
(!=
) - Is x
(mathematically) not equal to self
$contains(x)
(%inset%
) - Is x
contained in self
All methods are vectorized for multiple testing.
s = Set$new(1,2,3) s$contains(1) #> [1] TRUE s$contains(2, 4) #> [1] TRUE c(2, 4) %inset% s #> [1] TRUE FALSE s$isSubset(Set$new(1,2,3), proper = FALSE) #> [1] TRUE s$isSubset(Set$new(1,2,3), proper = TRUE) #> [1] FALSE c(Set$new(1), Set$new(4, 5)) < s #> [1] TRUE FALSE # Sets are FuzzySets with membership = 1 s$equals(FuzzySet$new(elements = 1:3, membership = 1)) #> [1] TRUE s$equals(FuzzySet$new(elements = 1:3, membership = 0.1)) #> [1] FALSE s == Set$new(1, 2, 3) #> [1] TRUE s != c(Set$new(1,2,3), Set$new(1, 2)) #> [1] FALSE TRUE 1:10 %inset% ConditionalSet$new(function(x) x %% 2 == 0) #> [1] FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE # The `bound` argument in `isSubset` is used for specifying # how open interval containedness should be checked i = Interval$new(1, 10, type = "(]") i$contains(Set$new(1), bound = FALSE) #> [1] FALSE i$contains(Set$new(10), bound = FALSE) #> [1] FALSE i$contains(Set$new(1), bound = TRUE) #> [1] FALSE i$contains(Set$new(10), bound = TRUE) #> [1] FALSE
set6
includes the following operations:
setunion
(+
) - Union of multiple setspowerset
- Powerset of a given setsetpower
(^
) - n-ary cartesian product of a given setsetcomplement
(-
) - Relative complement, or set difference, of two setssetintersect
(&
) - Intersection of two setssetproduct
(*
) - Cartesian product of multiple setssetsymdiff
(%-%
) - Symmetric difference of two setsWe will look at the most common of these below.
The union of sets is defined the as the set of elements in all the sets of interest.
Set$new(1) + Set$new(2) + Set$new(3) #> {1, 2, 3} Interval$new(1, 10) + Set$new(1) # no effect #> [1,10] setunion(Set$new(1,2), Interval$new(3, 10), Set$new(16)) #> {1,...,2} ∪ [3,10] ∪ {16,...,16} PosReals$new() + NegReals$new() #> ℝ
The relative complement of two sets, \(A-B\), is defined as the set of elements in \(A\) but not in \(B\).
Set$new(elements = 1:10) - Set$new(elements = 4:10) #> {1, 2, 3} Set$new(1,2,3,4) - Set$new(2) #> {1, 3, 4} Reals$new() - PosReals$new() #> ℝ- Interval$new(5, 10) - Interval$new(3, 12) #> ∅ Interval$new(5, 10) - Interval$new(7, 12) #> [5,7) Interval$new(5, 10) - Interval$new(11, 12) # no effect #> [5,10]
The cartesian product of multiple sets is often confused with the n-ary cartesian product, read the help page at ?setproduct
for a full description of the problem. Both forms are allowed in set6
with the nest
argument.
Set$new(1, 2) * Set$new(3, 4) #> {1, 2} × {3, 4} Set$new(1, 2) * Set$new(3, 4) * Set$new(5, 6) # n-ary #> {1, 2} × {3, 4} × {5, 6} # nest = FALSE default - we will return to the `simplify` argument below setproduct(Set$new(1, 2), Set$new(3, 4), Set$new(5, 6), nest = TRUE, simplify = TRUE) #> {((1, 3), 5), ((2, 3), 5),...,((1, 4), 6), ((2, 4), 6)} setproduct(Set$new(1, 2), Set$new(3, 4), Set$new(5, 6), nest = FALSE, simplify = TRUE) #> {(1, 3, 5), (2, 3, 5),...,(1, 4, 6), (2, 4, 6)} # n-ary cartesian product on the same set setpower(Set$new(1,2), 3, simplify = TRUE) #> {(1, 1, 1), (2, 1, 1),...,(1, 2, 2), (2, 2, 2)}
Finally we look briefly at wrappers, and the simplify
argument. Each operation has an associated wrapper that will be created if simplify == FALSE
or if the resulting set is too complicated to return as a single Set
object. The operations concerned with products, i.e. setproduct
, setpower
, powerset
, all have simplify == FALSE
as the default; whereas the others concerned with unions and differences, have simplify == TRUE
as the default.
# default: simplify = TRUE setunion(Set$new(1,2,3), Set$new(4,5)) #> {1, 2,...,4, 5} setunion(Set$new(1,2,3), Set$new(4,5), simplify = FALSE) #> {1, 2, 3} ∪ {4, 5} # default: simplify = FALSE setproduct(Set$new(1,2), Set$new(4,5)) #> {1, 2} × {4, 5} setproduct(Set$new(1,2), Set$new(4,5), simplify = TRUE) #> {(1, 4), (2, 4), (1, 5), (2, 5)} # default: simplify = FALSE powerset(Set$new(1,2,3)) #> ℘({1, 2, 3}) powerset(Set$new(1,2,3), simplify = TRUE) #> {∅, {1},...,{2, 3}, {1, 2, 3}}
All wrappers inherit from Set
and therefore share the same methods and fields.
set6
is still in its early stages and the API is considered to be maturing, this means that whilst no big changes are expected to the design, minor ones can be expected. set6
will be considered to be in its version 1 once all methods and fields are finalised (some are missing from wrappers), and when the code has been refactored to be more efficient. See the website for more tutorials and follow/star on GitHub for updates.