Introduction to set6

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.

Constructing a Set

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 contain any object in R that has a valid as.character dispatch method that ensures uniqueness in objects. This is a requirement as internally string representations are used to compare 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))
#> {0+2i, 1, a, {1}}

The different kind of sets

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 ∈ 𝕍 : x%%2 == 0}

# 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

Comparisons and Containedness

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

Algebra of Sets

set6 includes the following operations:

setunion (+) - Union of multiple sets
powerset - Powerset of a given set
setpower (^) - n-ary cartesian product of a given set
setcomplement (-) - Relative complement, or set difference, of two sets
setintersect (&) - Intersection of two sets
setproduct (*) - Cartesian product of multiple sets
setsymdiff (%-%) - Symmetric difference of two sets

We will look at the most common of these below.

Union of Sets

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

Relative Complement

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]

Cartesian Product

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$new(5, 6), nest = TRUE, simplify = TRUE)
#> {((1, 3), 5), ((1, 3), 6),...,((2, 4), 5), ((2, 4), 6)}
setproduct(Set$new(1, 2), Set$new(3, 4), Set$new(5, 6), nest = FALSE, simplify = TRUE)
#> {(1, 3, 5), (1, 3, 6),...,(2, 4, 5), (2, 4, 6)}

# n-ary cartesian product on the same set
setpower(Set$new(1,2), 3, simplify = TRUE)
#> {(1, 1, 1), (1, 1, 2),...,(2, 2, 1), (2, 2, 2)}

Intersection

The intersection of two sets is defined as the set of elements that lie in both sets.

Set$new(1,2,3) & Set$new(3,5,6)
#> {3}
Set$new(5,6) & Reals$new()
#> {5, 6}
Set$new(1) & Set$new(2)
#> ∅

Wrappers

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. Wrappers faciliate lazy evaluation and symbolic representation by providing unique character representations of all sets after operations and do not evaluate the set elements unless specifically requested. In general wrappers should not be directly constructed but instead only used as the result of operations. 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), (1, 5), (2, 4), (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.

u = setunion(Set$new(1,2,3), Set$new(4,5), simplify = FALSE)
c(2,5,8) %inset% u
#> [1]  TRUE  TRUE FALSE

p = Set$new(1,2) * Set$new(3,4)
p$contains(Tuple$new(2, 4))
#> [1] TRUE