Introduction to set6

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 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}}

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%%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

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.

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

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]

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)}

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. 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.

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

Going Forward

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.