Package 'set6'

Description

set6 upgrades the {sets} package to R6. Many forms of mathematical sets are implemented, including (countably finite) sets, tuples, intervals (countably infinite or uncountable), and fuzzy variants. Wrappers extend functionality by allowing symbolic representations of complex operations on sets, including unions, (cartesian) products, exponentiation, and differences (asymmetric and symmetric).

Details

The main features of set6 are:

• Object-oriented programming, which allows a clear inheritance structure for Sets, Intervals, Tuples, and other variants.

• Set operations and wrappers for both explicit and symbolic representations for algebra of sets.

• Methods for assertions and comparison checks, including subsets, equality, and containedness.

To learn more about set6, start with the set6 vignette:

vignette("set6", "set6")

And for more advanced usage see the complete tutorials at

https://github.com/xoopR/set6

Author(s)

Maintainer: Raphael Sonabend [email protected] (ORCID)

Authors:

• Franz Kiraly [email protected]

See Also

Useful links:

• https://xoopR.github.io/set6/

• https://github.com/xoopR/set6

• Report bugs at https://github.com/xoopR/set6/issues

Description

Coerces object to an R6 FuzzySet/FuzzyTuple

Usage

as.FuzzySet(object)

## S3 method for class 'numeric'
as.FuzzySet(object)

## S3 method for class 'list'
as.FuzzySet(object)

## S3 method for class 'matrix'
as.FuzzySet(object)

## S3 method for class 'data.frame'
as.FuzzySet(object)

## S3 method for class 'Set'
as.FuzzySet(object)

## S3 method for class 'FuzzySet'
as.FuzzySet(object)

## S3 method for class 'Interval'
as.FuzzySet(object)

## S3 method for class 'ConditionalSet'
as.FuzzySet(object)

as.FuzzyTuple(object)

## S3 method for class 'numeric'
as.FuzzyTuple(object)

## S3 method for class 'list'
as.FuzzyTuple(object)

## S3 method for class 'matrix'
as.FuzzyTuple(object)

## S3 method for class 'data.frame'
as.FuzzyTuple(object)

## S3 method for class 'Set'
as.FuzzyTuple(object)

## S3 method for class 'FuzzySet'
as.FuzzyTuple(object)

## S3 method for class 'Interval'
as.FuzzyTuple(object)

## S3 method for class 'ConditionalSet'
as.FuzzyTuple(object)

as.FuzzyMultiset(object)

## S3 method for class 'numeric'
as.FuzzyMultiset(object)

## S3 method for class 'list'
as.FuzzyMultiset(object)

## S3 method for class 'matrix'
as.FuzzyMultiset(object)

## S3 method for class 'data.frame'
as.FuzzyMultiset(object)

## S3 method for class 'Set'
as.FuzzyMultiset(object)

## S3 method for class 'FuzzySet'
as.FuzzyMultiset(object)

## S3 method for class 'Interval'
as.FuzzyMultiset(object)

## S3 method for class 'ConditionalSet'
as.FuzzyMultiset(object)

Arguments

Details

• as.FuzzySet.list - Assumes list has two items, named elements and membership, and that they are ordered to be corresponding.

• as.FuzzySet.matrix - Assumes first column corresponds to elements and second column corresponds to their respective membership.

• as.FuzzySet.data.frame - First checks to see if one column is called elements and the other is called membership. If not then uses as.FuzzySet.matrix.

• as.FuzzySet.Set - Creates a FuzzySet by assuming Set elements all have membership equal to $1$.

• as.FuzzySet.Interval - First tries coercion via as.Set.Interval then uses as.FuzzySet.Set.

See Also

FuzzySet FuzzyTuple

Other coercions: as.Interval(), as.Set()

Description

Coerces object to an R6 Interval.

Usage

as.Interval(object)

## S3 method for class 'Set'
as.Interval(object)

## S3 method for class 'Interval'
as.Interval(object)

## S3 method for class 'list'
as.Interval(object)

## S3 method for class 'data.frame'
as.Interval(object)

## S3 method for class 'matrix'
as.Interval(object)

## S3 method for class 'numeric'
as.Interval(object)

## S3 method for class 'ConditionalSet'
as.Interval(object)

Arguments

Details

• as.Interval.list/as.Interval.data.frame - Assumes the list/data.frame has named items/columns: ⁠lower, upper, type, class⁠.

• as.Interval.numeric - If the numeric vector is a continuous interval with no breaks then coerces to an Interval with: ⁠lower = min(object), upper = max(object), class = "integer"⁠. Ordering is ignored.

• as.Interval.matrix - Tries coercion via as.Interval.numeric on the first column of the matrix.

• as.Interval.Set - First tries coercion via as.Interval.numeric, if possible wraps result in a Set.

• as.Interval.FuzzySet - Tries coercion via as.Interval.Set on the support of the FuzzySet.

See Also

Interval

Other coercions: as.FuzzySet(), as.Set()

Description

Coerces object to an R6 Set/Tuple

Usage

as.Set(object)

## Default S3 method:
as.Set(object)

## S3 method for class 'numeric'
as.Set(object)

## S3 method for class 'list'
as.Set(object)

## S3 method for class 'matrix'
as.Set(object)

## S3 method for class 'data.frame'
as.Set(object)

## S3 method for class 'Set'
as.Set(object)

## S3 method for class 'FuzzySet'
as.Set(object)

## S3 method for class 'Interval'
as.Set(object)

## S3 method for class 'ConditionalSet'
as.Set(object)

as.Tuple(object)

## Default S3 method:
as.Tuple(object)

## S3 method for class 'numeric'
as.Tuple(object)

## S3 method for class 'list'
as.Tuple(object)

## S3 method for class 'matrix'
as.Tuple(object)

## S3 method for class 'data.frame'
as.Tuple(object)

## S3 method for class 'FuzzySet'
as.Tuple(object)

## S3 method for class 'Set'
as.Tuple(object)

## S3 method for class 'Interval'
as.Tuple(object)

## S3 method for class 'ConditionalSet'
as.Tuple(object)

as.Multiset(object)

## Default S3 method:
as.Multiset(object)

## S3 method for class 'numeric'
as.Multiset(object)

## S3 method for class 'list'
as.Multiset(object)

## S3 method for class 'matrix'
as.Multiset(object)

## S3 method for class 'data.frame'
as.Multiset(object)

## S3 method for class 'FuzzySet'
as.Multiset(object)

## S3 method for class 'Set'
as.Multiset(object)

## S3 method for class 'Interval'
as.Multiset(object)

## S3 method for class 'ConditionalSet'
as.Multiset(object)

Arguments

Details

• as.Set.default - Creates a Set using the object as the elements.

• as.Set.list - Creates a Set for each element in list.

• as.Set.matrix/as.Set.data.frame - Creates a Set for each column in matrix/data.frame.

• as.Set.FuzzySet - Creates a Set from the support of the FuzzySet.

• as.Set.Interval - If the interval has finite cardinality then creates a Set from the Interval elements.

See Also

Set Tuple

Other coercions: as.FuzzySet(), as.Interval()

Description

ComplementSet class for symbolic complement of mathematical sets.

Details

The purpose of this class is to provide a symbolic representation for the complement of sets that cannot be represented in a simpler class. Whilst this is not an abstract class, it is not recommended to construct this class directly but via the set operation methods.

Super classes

set6::Set -> set6::SetWrapper -> ComplementSet

Active bindings

Methods

Public methods

• ComplementSet$new()

• ComplementSet$strprint()

• ComplementSet$contains()

• ComplementSet$clone()

Method new()

Create a new ComplementSet object. It is not recommended to construct this class directly.

Usage

ComplementSet$new(addset, subtractset, lower = NULL, upper = NULL, type = NULL)

Arguments

Returns

A new ComplementSet object.

Method strprint()

Creates a printable representation of the object.

Usage

ComplementSet$strprint(n = 2)

Arguments

Returns

A character string representing the object.

Method contains()

Tests if elements x are contained in self.

Usage

ComplementSet$contains(x, all = FALSE, bound = FALSE)

Arguments

Returns

If all == TRUE then returns TRUE if all x are contained in self, otherwise FALSE. If all == FALSE returns a vector of logicals corresponding to the length of x, representing if each is contained in self. If bound == TRUE then an element is contained in self if it is on or within the (possibly-open) bounds of self, otherwise TRUE only if the element is within self or the bounds are closed.

Method clone()

The objects of this class are cloneable with this method.

Usage

ComplementSet$clone(deep = FALSE)

Arguments

See Also

Set operations: setunion, setproduct, setpower, setcomplement, setsymdiff, powerset, setintersect

Other wrappers: ExponentSet, PowersetSet, ProductSet, UnionSet

Description

The mathematical set of complex numbers, defined as the the set of reals with possibly imaginary components. i.e.

$\\{a + bi \\ : \\ a,b \in R\\}$

where $R$ is the set of reals.

Details

There is no inherent ordering in the set of complex numbers, hence only the contains method is implemented here.

Super class

set6::Set -> Complex

Methods

Public methods

• Complex$new()

• Complex$contains()

• Complex$equals()

• Complex$isSubset()

• Complex$strprint()

• Complex$clone()

Method new()

Create a new Complex object.

Usage

Complex$new()

Returns

A new Complex object.

Method contains()

Tests to see if x is contained in the Set.

Usage

Complex$contains(x, all = FALSE, bound = NULL)

Arguments

Details

x can be of any type, including a Set itself. x should be a tuple if checking to see if it lies within a set of dimension greater than one. To test for multiple x at the same time, then provide these as a list.

If all = TRUE then returns TRUE if all x are contained in the Set, otherwise returns a vector of logicals. For Intervals, bound is used to specify if elements lying on the (possibly open) boundary of the interval are considered contained (bound = TRUE) or not (bound = FALSE).

Returns

If all is TRUE then returns TRUE if all elements of x are contained in the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

The infix operator ⁠%inset%⁠ is available to test if x is an element in the Set, see examples.

Method equals()

Tests if two sets are equal.

Usage

Complex$equals(x, all = FALSE)

Arguments

Returns

If all is TRUE then returns TRUE if all x are equal to the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Infix operators can be used for:

Examples

# Equals
Set$new(1,2)$equals(Set$new(5,6))
Set$new(1,2)$equals(Interval$new(1,2))
Set$new(1,2) == Interval$new(1,2, class = "integer")

# Not equal
!Set$new(1,2)$equals(Set$new(1,2))
Set$new(1,2) != Set$new(1,5)

Method isSubset()

Test if one set is a (proper) subset of another

Usage

Complex$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Details

If using the method directly, and not via one of the operators then the additional boolean argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a (non-proper) subset if it is fully contained by, or equal to, the other Set.

When calling ⁠$isSubset⁠ on objects inheriting from Interval, the method treats the interval as if it is a Set, i.e. ordering and class are ignored. Use ⁠$isSubinterval⁠ to test if one interval is a subinterval of another.

Infix operators can be used for:

Every Set is a subset of a Universal. No Set is a super set of a Universal, and only a Universal is not a proper subset of a Universal.

Returns

If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Examples

Set$new(1,2,3)$isSubset(Set$new(1,2), proper = TRUE)
Set$new(1,2) < Set$new(1,2,3) # proper subset

c(Set$new(1,2,3), Set$new(1)) < Set$new(1,2,3) # not proper
Set$new(1,2,3) <= Set$new(1,2,3) # proper

Method strprint()

Creates a printable representation of the object.

Usage

Complex$strprint(n = 2)

Arguments

Returns

A character string representing the object.

Method clone()

The objects of this class are cloneable with this method.

Usage

Complex$clone(deep = FALSE)

Arguments

See Also

Other special sets: ExtendedReals, Integers, Logicals, Naturals, NegIntegers, NegRationals, NegReals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Examples

## ------------------------------------------------
## Method `Complex$equals`
## ------------------------------------------------

# Equals
Set$new(1,2)$equals(Set$new(5,6))
Set$new(1,2)$equals(Interval$new(1,2))
Set$new(1,2) == Interval$new(1,2, class = "integer")

# Not equal
!Set$new(1,2)$equals(Set$new(1,2))
Set$new(1,2) != Set$new(1,5)

## ------------------------------------------------
## Method `Complex$isSubset`
## ------------------------------------------------

Set$new(1,2,3)$isSubset(Set$new(1,2), proper = TRUE)
Set$new(1,2) < Set$new(1,2,3) # proper subset

c(Set$new(1,2,3), Set$new(1)) < Set$new(1,2,3) # not proper
Set$new(1,2,3) <= Set$new(1,2,3) # proper

Description

A mathematical set defined by one or more logical conditions.

Details

Conditional sets are a useful tool for symbolically defining possibly infinite sets. They can be combined using standard 'and', &, and 'or', |, operators.

Super class

set6::Set -> ConditionalSet

Active bindings

Methods

Public methods

• ConditionalSet$new()

• ConditionalSet$contains()

• ConditionalSet$equals()

• ConditionalSet$strprint()

• ConditionalSet$summary()

• ConditionalSet$isSubset()

• ConditionalSet$clone()

Method new()

Create a new ConditionalSet object.

Usage

ConditionalSet$new(condition = function(x) TRUE, argclass = NULL)

Arguments

Details

The condition should be given as a function that when evaluated returns either TRUE or FALSE. Further constraints can be given by providing the universe of the function arguments as Sets, if these are not given then Universal is assumed. See examples. Defaults construct the Universal set.

Returns

A new ConditionalSet object.

Method contains()

Tests to see if x is contained in the Set.

Usage

ConditionalSet$contains(x, all = FALSE, bound = NULL)

Arguments

Details

x can be of any type, including a Set itself. x should be a tuple if checking to see if it lies within a set of dimension greater than one. To test for multiple x at the same time, then provide these as a list.

If all = TRUE then returns TRUE if all x are contained in the Set, otherwise returns a vector of logicals.

An element is contained in a ConditionalSet if it returns TRUE as an argument in the defining function. For sets that are defined with a function that takes multiple arguments, a Tuple should be passed to x.

Returns

If all is TRUE then returns TRUE if all elements of x are contained in the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

The infix operator ⁠%inset%⁠ is available to test if x is an element in the Set, see examples.

Examples

# Set of positives
s = ConditionalSet$new(function(x) x > 0)
s$contains(list(1,-1))

# Set via equality
s = ConditionalSet$new(function(x, y) x + y == 2)
s$contains(list(Set$new(2, 0), Set$new(0, 2)))

# Tuples are recommended when using contains as they allow non-unique elements
s = ConditionalSet$new(function(x, y) x + y == 4)
\dontrun{
s$contains(Set$new(2, 2)) # Errors as Set$new(2,2) == Set$new(2)
}

# Set of Positive Naturals
s = ConditionalSet$new(function(x) TRUE, argclass = list(x = PosNaturals$new()))
s$contains(list(-2, 2))

Method equals()

Tests if two sets are equal.

Usage

ConditionalSet$equals(x, all = FALSE)

Arguments

Details

Two sets are equal if they contain the same elements. Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are equal to the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Method strprint()

Creates a printable representation of the object.

Usage

ConditionalSet$strprint(n = NULL)

Arguments

Returns

A character string representing the object.

Method summary()

See strprint.

Usage

ConditionalSet$summary(n = NULL)

Arguments

Method isSubset()

Currently undefined for ConditionalSets.

Usage

ConditionalSet$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

ConditionalSet$clone(deep = FALSE)

Arguments

See Also

Other sets: FuzzyMultiset, FuzzySet, FuzzyTuple, Interval, Multiset, Set, Tuple

Examples

# Set of Positive Naturals
s <- ConditionalSet$new(function(x) TRUE, argclass = list(x = PosNaturals$new()))

## ------------------------------------------------
## Method `ConditionalSet$contains`
## ------------------------------------------------

# Set of positives
s = ConditionalSet$new(function(x) x > 0)
s$contains(list(1,-1))

# Set via equality
s = ConditionalSet$new(function(x, y) x + y == 2)
s$contains(list(Set$new(2, 0), Set$new(0, 2)))

# Tuples are recommended when using contains as they allow non-unique elements
s = ConditionalSet$new(function(x, y) x + y == 4)
## Not run: 
s$contains(Set$new(2, 2)) # Errors as Set$new(2,2) == Set$new(2)

## End(Not run)

# Set of Positive Naturals
s = ConditionalSet$new(function(x) TRUE, argclass = list(x = PosNaturals$new()))
s$contains(list(-2, 2))

Description

Operator for ⁠$contains⁠ methods. See Set⁠$contains⁠ for full details. Operators can be used for:

Usage

x %inset% y

Arguments

Description

Operator for ⁠$equals⁠ methods. See Set⁠$equals⁠ for full details. Operators can be used for:

Usage

## S3 method for class 'Set'
x == y

## S3 method for class 'Set'
x != y

Arguments

Description

ExponentSet class for symbolic exponentiation of mathematical sets.

Details

The purpose of this class is to provide a symbolic representation for the exponentiation of sets that cannot be represented in a simpler class. Whilst this is not an abstract class, it is not recommended to construct this class directly but via the set operation methods.

Super classes

set6::Set -> set6::SetWrapper -> set6::ProductSet -> ExponentSet

Active bindings

Methods

Public methods

• ExponentSet$new()

• ExponentSet$strprint()

• ExponentSet$contains()

• ExponentSet$clone()

Method new()

Create a new ExponentSet object. It is not recommended to construct this class directly.

Usage

ExponentSet$new(set, power)

Arguments

Returns

A new ExponentSet object.

Method strprint()

Creates a printable representation of the object.

Usage

ExponentSet$strprint(n = 2)

Arguments

Returns

A character string representing the object.

Method contains()

Tests if elements x are contained in self.

Usage

ExponentSet$contains(x, all = FALSE, bound = FALSE)

Arguments

Returns

If all == TRUE then returns TRUE if all x are contained in self, otherwise FALSE. If all == FALSE returns a vector of logicals corresponding to the length of x, representing if each is contained in self. If bound == TRUE then an element is contained in self if it is on or within the (possibly-open) bounds of self, otherwise TRUE only if the element is within self or the bounds are closed.

Method clone()

The objects of this class are cloneable with this method.

Usage

ExponentSet$clone(deep = FALSE)

Arguments

See Also

Set operations: setunion, setproduct, setpower, setcomplement, setsymdiff, powerset, setintersect

Other wrappers: ComplementSet, PowersetSet, ProductSet, UnionSet

Description

The mathematical set of extended real numbers, defined as the union of the set of reals with $\pm\infty$. i.e.

$R \cup \\{-\infty, \infty\\}$

where $R$ is the set of reals.

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Reals -> ExtendedReals

Methods

Public methods

• ExtendedReals$new()

• ExtendedReals$clone()

Method new()

Create a new ExtendedReals object.

Usage

ExtendedReals$new()

Returns

A new ExtendedReals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

ExtendedReals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, Integers, Logicals, Naturals, NegIntegers, NegRationals, NegReals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Description

A general FuzzyMultiset object for mathematical fuzzy multisets, inheriting from FuzzySet.

Details

Fuzzy multisets generalise standard mathematical multisets to allow for fuzzy relationships. Whereas a standard, or crisp, multiset assumes that an element is either in a multiset or not, a fuzzy multiset allows an element to be in a multiset to a particular degree, known as the membership function, which quantifies the inclusion of an element by a number in [0, 1]. Thus a (crisp) multiset is a fuzzy multiset where all elements have a membership equal to $1$. Similarly to Multisets, elements do not need to be unique.

Super classes

set6::Set -> set6::FuzzySet -> FuzzyMultiset

Methods

Public methods

• FuzzyMultiset$equals()

• FuzzyMultiset$isSubset()

• FuzzyMultiset$alphaCut()

• FuzzyMultiset$clone()

Method equals()

Tests if two sets are equal.

Usage

FuzzyMultiset$equals(x, all = FALSE)

Arguments

Details

Two fuzzy sets are equal if they contain the same elements with the same memberships and in the same order. Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are equal to the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Method isSubset()

Test if one set is a (proper) subset of another

Usage

FuzzyMultiset$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Details

If using the method directly, and not via one of the operators then the additional boolean argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a (non-proper) subset if it is fully contained by, or equal to, the other Set.

Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Method alphaCut()

The alpha-cut of a fuzzy set is defined as the set

$A_\alpha = \{x \epsilon F | m \ge \alpha\}$

where $x$ is an element in the fuzzy set, $F$, and $m$ is the corresponding membership.

Usage

FuzzyMultiset$alphaCut(alpha, strong = FALSE, create = FALSE)

Arguments

Returns

Elements in FuzzyMultiset or a Set of the elements.

Method clone()

The objects of this class are cloneable with this method.

Usage

FuzzyMultiset$clone(deep = FALSE)

Arguments

See Also

Other sets: ConditionalSet, FuzzySet, FuzzyTuple, Interval, Multiset, Set, Tuple

Examples

# Different constructors
FuzzyMultiset$new(1, 0.5, 2, 1, 3, 0)
FuzzyMultiset$new(elements = 1:3, membership = c(0.5, 1, 0))

# Crisp sets are a special case FuzzyMultiset
# Note membership defaults to full membership
FuzzyMultiset$new(elements = 1:5) == Multiset$new(1:5)

f <- FuzzyMultiset$new(1, 0.2, 2, 1, 3, 0)
f$membership()
f$alphaCut(0.3)
f$core()
f$inclusion(0)
f$membership(0)
f$membership(1)

# Elements can be duplicated, and with different memberships,
#  although this is not necessarily sensible.
FuzzyMultiset$new(1, 0.1, 1, 1)

# Like FuzzySets, ordering does not matter.
FuzzyMultiset$new(1, 0.1, 2, 0.2) == FuzzyMultiset$new(2, 0.2, 1, 0.1)

Description

A general FuzzySet object for mathematical fuzzy sets, inheriting from Set.

Details

Fuzzy sets generalise standard mathematical sets to allow for fuzzy relationships. Whereas a standard, or crisp, set assumes that an element is either in a set or not, a fuzzy set allows an element to be in a set to a particular degree, known as the membership function, which quantifies the inclusion of an element by a number in [0, 1]. Thus a (crisp) set is a fuzzy set where all elements have a membership equal to $1$. Similarly to Sets, elements must be unique and the ordering does not matter, to establish order and non-unique elements, FuzzyTuples can be used.

Super class

set6::Set -> FuzzySet

Methods

Public methods

• FuzzySet$new()

• FuzzySet$strprint()

• FuzzySet$membership()

• FuzzySet$alphaCut()

• FuzzySet$support()

• FuzzySet$core()

• FuzzySet$inclusion()

• FuzzySet$equals()

• FuzzySet$isSubset()

• FuzzySet$clone()

Method new()

Create a new FuzzySet object.

Usage

FuzzySet$new(
  ...,
  elements = NULL,
  membership = rep(1, length(elements)),
  class = NULL
)

Arguments

Details

FuzzySets can be constructed in one of two ways, either by supplying the elements and their membership in alternate order, or by providing a list of elements to elements and a list of respective memberships to membership, see examples. If the class argument is non-NULL, then all elements will be coerced to the given class in construction, and if elements of a different class are added these will either be rejected or coerced.

Returns

A new FuzzySet object.

Method strprint()

Creates a printable representation of the object.

Usage

FuzzySet$strprint(n = 2)

Arguments

Returns

A character string representing the object.

Method membership()

Returns the membership, i.e. value in [0, 1], of either the given element(s) or all elements in the fuzzy set.

Usage

FuzzySet$membership(element = NULL)

Arguments

Details

For FuzzySets this is straightforward and returns the membership of the given element(s), however in FuzzyTuples and FuzzyMultisets when an element may be duplicated, the function returns the membership of all instances of the element.

Returns

Value, or list of values, in [0, 1].

Examples

f = FuzzySet$new(1, 0.1, 2, 0.5, 3, 1)
f$membership()
f$membership(2)
f$membership(list(1, 2))

Method alphaCut()

The alpha-cut of a fuzzy set is defined as the set

$A_\alpha = \{x \epsilon F | m \ge \alpha\}$

where $x$ is an element in the fuzzy set, $F$, and $m$ is the corresponding membership.

Usage

FuzzySet$alphaCut(alpha, strong = FALSE, create = FALSE)

Arguments

Returns

Elements in FuzzySet or a Set of the elements.

Examples

f = FuzzySet$new(1, 0.1, 2, 0.5, 3, 1)
# Alpha-cut
f$alphaCut(0.5)

# Strong alpha-cut
f$alphaCut(0.5, strong = TRUE)

# Create a set from the alpha-cut
f$alphaCut(0.5, create = TRUE)

Method support()

The support of a fuzzy set is defined as the set of elements whose membership is greater than zero, or the strong alpha-cut with $\alpha = 0$,

$A_\alpha = \{x \epsilon F | m > 0\}$

where $x$ is an element in the fuzzy set, $F$, and $m$ is the corresponding membership.

Usage

FuzzySet$support(create = FALSE)

Arguments

Returns

Support elements in fuzzy set or a Set of the support elements.

Examples

f = FuzzySet$new(0.1, 0, 1, 0.1, 2, 0.5, 3, 1)
f$support()
f$support(TRUE)

Method core()

The core of a fuzzy set is defined as the set of elements whose membership is equal to one, or the alpha-cut with $\alpha = 1$,

$A_\alpha = \{x \epsilon F \ : \ m \ge 1\}$

where $x$ is an element in the fuzzy set, $F$, and $m$ is the corresponding membership.

Usage

FuzzySet$core(create = FALSE)

Arguments

Returns

Core elements in FuzzySet or a Set of the core elements.

Examples

f = FuzzySet$new(0.1, 0, 1, 0.1, 2, 0.5, 3, 1)
f$core()
f$core(TRUE)

Method inclusion()

An element in a fuzzy set, with corresponding membership $m$, is:

• Included - If $m = 1$

• Partially Included - If $0 < m < 1$

• Not Included - If $m = 0$

Usage

FuzzySet$inclusion(element)

Arguments

Details

For FuzzySets this is straightforward and returns the inclusion level of the given element(s), however in FuzzyTuples and FuzzyMultisets when an element may be duplicated, the function returns the inclusion level of all instances of the element.

Returns

One of: "Included", "Partially Included", "Not Included"

Examples

f = FuzzySet$new(0.1, 0, 1, 0.1, 2, 0.5, 3, 1)
f$inclusion(0.1)
f$inclusion(1)
f$inclusion(3)

Method equals()

Tests if two sets are equal.

Usage

FuzzySet$equals(x, all = FALSE)

Arguments

Details

Two fuzzy sets are equal if they contain the same elements with the same memberships. Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are equal to the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Method isSubset()

Test if one set is a (proper) subset of another

Usage

FuzzySet$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Details

If using the method directly, and not via one of the operators then the additional boolean argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a (non-proper) subset if it is fully contained by, or equal to, the other Set.

Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Method clone()

The objects of this class are cloneable with this method.

Usage

FuzzySet$clone(deep = FALSE)

Arguments

See Also

Other sets: ConditionalSet, FuzzyMultiset, FuzzyTuple, Interval, Multiset, Set, Tuple

Examples

# Different constructors
FuzzySet$new(1, 0.5, 2, 1, 3, 0)
FuzzySet$new(elements = 1:3, membership = c(0.5, 1, 0))

# Crisp sets are a special case FuzzySet
# Note membership defaults to full membership
FuzzySet$new(elements = 1:5) == Set$new(1:5)

f <- FuzzySet$new(1, 0.2, 2, 1, 3, 0)
f$membership()
f$alphaCut(0.3)
f$core()
f$inclusion(0)
f$membership(0)
f$membership(1)

## ------------------------------------------------
## Method `FuzzySet$membership`
## ------------------------------------------------

f = FuzzySet$new(1, 0.1, 2, 0.5, 3, 1)
f$membership()
f$membership(2)
f$membership(list(1, 2))

## ------------------------------------------------
## Method `FuzzySet$alphaCut`
## ------------------------------------------------

f = FuzzySet$new(1, 0.1, 2, 0.5, 3, 1)
# Alpha-cut
f$alphaCut(0.5)

# Strong alpha-cut
f$alphaCut(0.5, strong = TRUE)

# Create a set from the alpha-cut
f$alphaCut(0.5, create = TRUE)

## ------------------------------------------------
## Method `FuzzySet$support`
## ------------------------------------------------

f = FuzzySet$new(0.1, 0, 1, 0.1, 2, 0.5, 3, 1)
f$support()
f$support(TRUE)

## ------------------------------------------------
## Method `FuzzySet$core`
## ------------------------------------------------

f = FuzzySet$new(0.1, 0, 1, 0.1, 2, 0.5, 3, 1)
f$core()
f$core(TRUE)

## ------------------------------------------------
## Method `FuzzySet$inclusion`
## ------------------------------------------------

f = FuzzySet$new(0.1, 0, 1, 0.1, 2, 0.5, 3, 1)
f$inclusion(0.1)
f$inclusion(1)
f$inclusion(3)

Description

A general FuzzyTuple object for mathematical fuzzy tuples, inheriting from FuzzySet.

Details

Fuzzy tuples generalise standard mathematical tuples to allow for fuzzy relationships. Whereas a standard, or crisp, tuple assumes that an element is either in a tuple or not, a fuzzy tuple allows an element to be in a tuple to a particular degree, known as the membership function, which quantifies the inclusion of an element by a number in [0, 1]. Thus a (crisp) tuple is a fuzzy tuple where all elements have a membership equal to $1$. Similarly to Tuples, elements do not need to be unique and the ordering does matter, FuzzySets are special cases where the ordering does not matter and elements must be unique.

Super classes

set6::Set -> set6::FuzzySet -> FuzzyTuple

Methods

Public methods

• FuzzyTuple$equals()

• FuzzyTuple$isSubset()

• FuzzyTuple$alphaCut()

• FuzzyTuple$clone()

Method equals()

Tests if two sets are equal.

Usage

FuzzyTuple$equals(x, all = FALSE)

Arguments

Details

Two fuzzy sets are equal if they contain the same elements with the same memberships and in the same order. Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are equal to the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Method isSubset()

Test if one set is a (proper) subset of another

Usage

FuzzyTuple$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Details

If using the method directly, and not via one of the operators then the additional boolean argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a (non-proper) subset if it is fully contained by, or equal to, the other Set.

Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Method alphaCut()

The alpha-cut of a fuzzy set is defined as the set

$A_\alpha = \{x \epsilon F | m \ge \alpha\}$

where $x$ is an element in the fuzzy set, $F$, and $m$ is the corresponding membership.

Usage

FuzzyTuple$alphaCut(alpha, strong = FALSE, create = FALSE)

Arguments

Returns

Elements in FuzzyTuple or a Set of the elements.

Method clone()

The objects of this class are cloneable with this method.

Usage

FuzzyTuple$clone(deep = FALSE)

Arguments

See Also

Other sets: ConditionalSet, FuzzyMultiset, FuzzySet, Interval, Multiset, Set, Tuple

Examples

# Different constructors
FuzzyTuple$new(1, 0.5, 2, 1, 3, 0)
FuzzyTuple$new(elements = 1:3, membership = c(0.5, 1, 0))

# Crisp sets are a special case FuzzyTuple
# Note membership defaults to full membership
FuzzyTuple$new(elements = 1:5) == Tuple$new(1:5)

f <- FuzzyTuple$new(1, 0.2, 2, 1, 3, 0)
f$membership()
f$alphaCut(0.3)
f$core()
f$inclusion(0)
f$membership(0)
f$membership(1)

# Elements can be duplicated, and with different memberships,
#  although this is not necessarily sensible.
FuzzyTuple$new(1, 0.1, 1, 1)

# More important is ordering.
FuzzyTuple$new(1, 0.1, 2, 0.2) != FuzzyTuple$new(2, 0.2, 1, 0.1)
FuzzySet$new(1, 0.1, 2, 0.2) == FuzzySet$new(2, 0.2, 1, 0.1)

Description

The mathematical set of integers, defined as the set of whole numbers. i.e.

$\\{...,-3, -2, -1, 0, 1, 2, 3,...\\}$

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> Integers

Methods

Public methods

• Integers$new()

• Integers$clone()

Method new()

Create a new Integers object.

Usage

Integers$new(lower = -Inf, upper = Inf, type = "()")

Arguments

Returns

A new Integers object.

Method clone()

The objects of this class are cloneable with this method.

Usage

Integers$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Logicals, Naturals, NegIntegers, NegRationals, NegReals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Description

A general Interval object for mathematical intervals, inheriting from Set. Intervals may be open, closed, or half-open; as well as bounded above, below, or not at all.

Details

The Interval class can be used for finite or infinite intervals, but often Sets will be preferred for integer intervals over a finite continuous range.

Super class

set6::Set -> Interval

Active bindings

Methods

Public methods

• Interval$new()

• Interval$strprint()

• Interval$equals()

• Interval$contains()

• Interval$isSubset()

• Interval$isSubinterval()

• Interval$clone()

Method new()

Create a new Interval object.

Usage

Interval$new(
  lower = -Inf,
  upper = Inf,
  type = c("[]", "(]", "[)", "()"),
  class = "numeric",
  universe = ExtendedReals$new()
)

Arguments

Details

Intervals are constructed by specifying the Interval limits, the boundary type, the class, and the possible universe. The universe differs from class as it is primarily used for the setcomplement method. Whereas class specifies if the interval takes integers or numerics, the universe specifies what range the interval could take.

Returns

A new Interval object.

Method strprint()

Creates a printable representation of the object.

Usage

Interval$strprint(...)

Arguments

Returns

A character string representing the object.

Method equals()

Tests if two sets are equal.

Usage

Interval$equals(x, all = FALSE)

Arguments

Details

Two Intervals are equal if they have the same: class, type, and bounds. Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are equal to the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Examples

Interval$new(1,5) == Interval$new(1,5)
Interval$new(1,5, class = "integer") != Interval$new(1,5,class="numeric")

Method contains()

Tests to see if x is contained in the Set.

Usage

Interval$contains(x, all = FALSE, bound = FALSE)

Arguments

Details

x can be of any type, including a Set itself. x should be a tuple if checking to see if it lies within a set of dimension greater than one. To test for multiple x at the same time, then provide these as a list.

If all = TRUE then returns TRUE if all x are contained in the Set, otherwise returns a vector of logicals. For Intervals, bound is used to specify if elements lying on the (possibly open) boundary of the interval are considered contained (bound = TRUE) or not (bound = FALSE).

Returns

If all is TRUE then returns TRUE if all elements of x are contained in the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

The infix operator ⁠%inset%⁠ is available to test if x is an element in the Set, see examples.

Examples

s = Set$new(1:5)

# Simplest case
s$contains(4)
8 %inset% s

# Test if multiple elements lie in the set
s$contains(4:6, all = FALSE)
s$contains(4:6, all = TRUE)

# Check if a tuple lies in a Set of higher dimension
s2 = s * s
s2$contains(Tuple$new(2,1))
c(Tuple$new(2,1), Tuple$new(1,7), 2) %inset% s2

Method isSubset()

Test if one set is a (proper) subset of another

Usage

Interval$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Details

If using the method directly, and not via one of the operators then the additional boolean argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a (non-proper) subset if it is fully contained by, or equal to, the other Set.

When calling isSubset on objects inheriting from Interval, the method treats the interval as if it is a Set, i.e. ordering and class are ignored. Use isSubinterval to test if one interval is a subinterval of another.

Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Examples

Interval$new(1,3) < Interval$new(1,5)
Set$new(1,3) < Interval$new(0,5)

Method isSubinterval()

Test if one interval is a (proper) subinterval of another

Usage

Interval$isSubinterval(x, proper = FALSE, all = FALSE)

Arguments

Details

If x is a Set then will be coerced to an Interval if possible. ⁠$isSubinterval⁠ differs from ⁠$isSubset⁠ in that ordering and class are respected in ⁠$isSubinterval⁠. See examples for a clearer illustration of the difference.

Returns

If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Examples

Interval$new(1,3)$isSubset(Set$new(1,2)) # TRUE
Interval$new(1,3)$isSubset(Set$new(2, 1)) # TRUE
Interval$new(1,3, class = "integer")$isSubinterval(Set$new(1, 2)) # TRUE
Interval$new(1,3)$isSubinterval(Set$new(1, 2)) # FALSE
Interval$new(1,3)$isSubinterval(Set$new(2, 1)) # FALSE

Reals$new()$isSubset(Integers$new()) # TRUE
Reals$new()$isSubinterval(Integers$new()) # FALSE

Method clone()

The objects of this class are cloneable with this method.

Usage

Interval$clone(deep = FALSE)

Arguments

See Also

Other sets: ConditionalSet, FuzzyMultiset, FuzzySet, FuzzyTuple, Multiset, Set, Tuple

Examples

# Set of Reals
Interval$new()

# Set of Integers
Interval$new(class = "integer")

# Half-open interval
i <- Interval$new(1, 10, "(]")
i$contains(c(1, 10))
i$contains(c(1, 10), bound = TRUE)

# Equivalent Set and Interval
Set$new(1:5) == Interval$new(1, 5, class = "integer")

# SpecialSets can provide more efficient implementation
Interval$new() == ExtendedReals$new()
Interval$new(class = "integer", type = "()") == Integers$new()

## ------------------------------------------------
## Method `Interval$equals`
## ------------------------------------------------

Interval$new(1,5) == Interval$new(1,5)
Interval$new(1,5, class = "integer") != Interval$new(1,5,class="numeric")

## ------------------------------------------------
## Method `Interval$contains`
## ------------------------------------------------

s = Set$new(1:5)

# Simplest case
s$contains(4)
8 %inset% s

# Test if multiple elements lie in the set
s$contains(4:6, all = FALSE)
s$contains(4:6, all = TRUE)

# Check if a tuple lies in a Set of higher dimension
s2 = s * s
s2$contains(Tuple$new(2,1))
c(Tuple$new(2,1), Tuple$new(1,7), 2) %inset% s2

## ------------------------------------------------
## Method `Interval$isSubset`
## ------------------------------------------------

Interval$new(1,3) < Interval$new(1,5)
Set$new(1,3) < Interval$new(0,5)

## ------------------------------------------------
## Method `Interval$isSubinterval`
## ------------------------------------------------

Interval$new(1,3)$isSubset(Set$new(1,2)) # TRUE
Interval$new(1,3)$isSubset(Set$new(2, 1)) # TRUE
Interval$new(1,3, class = "integer")$isSubinterval(Set$new(1, 2)) # TRUE
Interval$new(1,3)$isSubinterval(Set$new(1, 2)) # FALSE
Interval$new(1,3)$isSubinterval(Set$new(2, 1)) # FALSE

Reals$new()$isSubset(Integers$new()) # TRUE
Reals$new()$isSubinterval(Integers$new()) # FALSE

Description

Operator for ⁠$isSubset⁠ methods. See Set⁠$isSubset⁠ for full details. Operators can be used for:

Usage

## S3 method for class 'Set'
x < y

## S3 method for class 'Set'
x <= y

## S3 method for class 'Set'
x > y

## S3 method for class 'Set'
x >= y

Arguments

Description

Lists special sets that can be used in Set.

Usage

listSpecialSets(simplify = FALSE)

Arguments

Value

Either a list of characters (if simplify is TRUE) or a data.frame of SpecialSets and their traits.

Examples

listSpecialSets()
listSpecialSets(TRUE)

Description

The Logicals is defined as the Set containing the elements TRUE and FALSE.

Super class

set6::Set -> Logicals

Methods

Public methods

• Logicals$new()

• Logicals$clone()

Method new()

Create a new Logicals object.

Usage

Logicals$new()

Details

The Logical set is the set containing TRUE and FALSE.

Returns

A new Logicals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

Logicals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Naturals, NegIntegers, NegRationals, NegReals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Examples

l <- Logicals$new()
print(l)
l$contains(list(TRUE, 1, FALSE))

Description

The LogicalSet is defined as the Set containing the elements TRUE and FALSE.

Super class

set6::Set -> LogicalSet

Methods

Public methods

• LogicalSet$new()

• LogicalSet$clone()

Method new()

Create a new LogicalSet object.

Usage

LogicalSet$new()

Details

The Logical set is the set containing TRUE and FALSE.

Returns

A new LogicalSet object.

Method clone()

The objects of this class are cloneable with this method.

Usage

LogicalSet$clone(deep = FALSE)

Arguments

Examples

l <- LogicalSet$new()
print(l)
l$contains(list(TRUE, 1, FALSE))

Description

A general Multiset object for mathematical Multisets, inheriting from Set.

Details

Multisets are generalisations of sets that allow an element to be repeated. They can be thought of as Tuples without ordering.

Super class

set6::Set -> Multiset

Methods

Public methods

• Multiset$equals()

• Multiset$isSubset()

• Multiset$clone()

Method equals()

Tests if two sets are equal.

Usage

Multiset$equals(x, all = FALSE)

Arguments

Details

An object is equal to a Multiset if it contains all the same elements, and in the same order. Infix operators can be used for:

Returns

If all is TRUE then returns TRUE if all x are equal to the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Examples

Multiset$new(1,2) ==  Multiset$new(1,2)
Multiset$new(1,2) != Multiset$new(1,2)
Multiset$new(1,1) != Set$new(1,1)

Method isSubset()

Test if one set is a (proper) subset of another

Usage

Multiset$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Details

If using the method directly, and not via one of the operators then the additional boolean argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a (non-proper) subset if it is fully contained by, or equal to, the other Set.

When calling ⁠$isSubset⁠ on objects inheriting from Interval, the method treats the interval as if it is a Set, i.e. ordering and class are ignored. Use ⁠$isSubinterval⁠ to test if one interval is a subinterval of another.

Infix operators can be used for:

An object is a (proper) subset of a Multiset if it contains all (some) of the same elements, and in the same order.

Returns

If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual element of x.

Examples

Multiset$new(1,2,3) < Multiset$new(1,2,3,4)
Multiset$new(1,3,2) < Multiset$new(1,2,3,4)
Multiset$new(1,3,2,4) <= Multiset$new(1,2,3,4)

Method clone()

The objects of this class are cloneable with this method.

Usage

Multiset$clone(deep = FALSE)

Arguments

See Also

Other sets: ConditionalSet, FuzzyMultiset, FuzzySet, FuzzyTuple, Interval, Set, Tuple

Examples

# Multiset of integers
Multiset$new(1:5)

# Multiset of multiple types
Multiset$new("a", 5, Set$new(1), Multiset$new(2))

# Each Multiset has properties and traits
t <- Multiset$new(1, 2, 3)
t$traits
t$properties

# Elements can be duplicated
Multiset$new(2, 2) != Multiset$new(2)

# Ordering does not matter
Multiset$new(1, 2) == Multiset$new(2, 1)

## ------------------------------------------------
## Method `Multiset$equals`
## ------------------------------------------------

Multiset$new(1,2) ==  Multiset$new(1,2)
Multiset$new(1,2) != Multiset$new(1,2)
Multiset$new(1,1) != Set$new(1,1)

## ------------------------------------------------
## Method `Multiset$isSubset`
## ------------------------------------------------

Multiset$new(1,2,3) < Multiset$new(1,2,3,4)
Multiset$new(1,3,2) < Multiset$new(1,2,3,4)
Multiset$new(1,3,2,4) <= Multiset$new(1,2,3,4)

Description

The mathematical set of natural numbers, defined as the counting numbers. i.e.

$\\{0, 1, 2,...\\}$

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> Naturals

Methods

Public methods

• Naturals$new()

• Naturals$clone()

Method new()

Create a new Naturals object.

Usage

Naturals$new(lower = 0)

Arguments

Returns

A new Naturals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

Naturals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, NegIntegers, NegRationals, NegReals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Description

The mathematical set of negative integers, defined as the set of negative whole numbers. i.e.

$\\{...,-3, -2, -1, 0\\}$

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Integers -> NegIntegers

Methods

Public methods

• NegIntegers$new()

• NegIntegers$clone()

Method new()

Create a new NegIntegers object.

Usage

NegIntegers$new(zero = FALSE)

Arguments

Returns

A new NegIntegers object.

Method clone()

The objects of this class are cloneable with this method.

Usage

NegIntegers$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, Naturals, NegRationals, NegReals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Description

The mathematical set of negative rational numbers, defined as the set of numbers that can be written as a fraction of two integers and are non-positive. i.e.

$\\{\frac{p}{q} \ : \ p,q \ \in \ Z, \ p/q \le 0, \ q \ne 0\\}$

where $Z$ is the set of integers.

Details

The ⁠$contains⁠ method does not work for the set of Rationals as it is notoriously difficult/impossible to find an algorithm for determining if any given number is rational or not. Furthermore, computers must truncate all irrational numbers to rational numbers.

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Rationals -> NegRationals

Methods

Public methods

• NegRationals$new()

• NegRationals$clone()

Method new()

Create a new NegRationals object.

Usage

NegRationals$new(zero = FALSE)

Arguments

Returns

A new NegRationals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

NegRationals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, Naturals, NegIntegers, NegReals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Description

The mathematical set of negative real numbers, defined as the union of the set of negative rationals and negative irrationals. i.e.

$I^- \cup Q^-$

where $I^-$ is the set of negative irrationals and $Q^-$ is the set of negative rationals.

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Reals -> NegReals

Methods

Public methods

• NegReals$new()

• NegReals$clone()

Method new()

Create a new NegReals object.

Usage

NegReals$new(zero = FALSE)

Arguments

Returns

A new NegReals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

NegReals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, Naturals, NegIntegers, NegRationals, PosIntegers, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Description

The mathematical set of positive integers, defined as the set of positive whole numbers. i.e.

$\\{0, 1, 2, 3,...\\}$

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Integers -> PosIntegers

Methods

Public methods

• PosIntegers$new()

• PosIntegers$clone()

Method new()

Create a new PosIntegers object.

Usage

PosIntegers$new(zero = FALSE)

Arguments

Returns

A new PosIntegers object.

Method clone()

The objects of this class are cloneable with this method.

Usage

PosIntegers$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, Naturals, NegIntegers, NegRationals, NegReals, PosNaturals, PosRationals, PosReals, Rationals, Reals, Universal

Description

The mathematical set of positive natural numbers, defined as the positive counting numbers. i.e.

$\\{1, 2, 3,...\\}$

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Naturals -> PosNaturals

Methods

Public methods

• PosNaturals$new()

• PosNaturals$clone()

Method new()

Create a new PosNaturals object.

Usage

PosNaturals$new()

Returns

A new PosNaturals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

PosNaturals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, Naturals, NegIntegers, NegRationals, NegReals, PosIntegers, PosRationals, PosReals, Rationals, Reals, Universal

Description

The mathematical set of positive rational numbers, defined as the set of numbers that can be written as a fraction of two integers and are non-negative. i.e.

$\\{\frac{p}{q} \ : \ p,q \ \in \ Z, \ p/q \ge 0, \ q \ne 0\\}$

where $Z$ is the set of integers.

Details

The ⁠$contains⁠ method does not work for the set of Rationals as it is notoriously difficult/impossible to find an algorithm for determining if any given number is rational or not. Furthermore, computers must truncate all irrational numbers to rational numbers.

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Rationals -> PosRationals

Methods

Public methods

• PosRationals$new()

• PosRationals$clone()

Method new()

Create a new PosRationals object.

Usage

PosRationals$new(zero = FALSE)

Arguments

Returns

A new PosRationals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

PosRationals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, Naturals, NegIntegers, NegRationals, NegReals, PosIntegers, PosNaturals, PosReals, Rationals, Reals, Universal

Description

The mathematical set of positive real numbers, defined as the union of the set of positive rationals and positive irrationals. i.e.

$I^+ \cup Q^+$

where $I^+$ is the set of positive irrationals and $Q^+$ is the set of positive rationals.

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> set6::Reals -> PosReals

Methods

Public methods

• PosReals$new()

• PosReals$clone()

Method new()

Create a new PosReals object.

Usage

PosReals$new(zero = FALSE)

Arguments

Returns

A new PosReals object.

Method clone()

The objects of this class are cloneable with this method.

Usage

PosReals$clone(deep = FALSE)

Arguments

See Also

Other special sets: Complex, ExtendedReals, Integers, Logicals, Naturals, NegIntegers, NegRationals, NegReals, PosIntegers, PosNaturals, PosRationals, Rationals, Reals, Universal

Description

Calculates and returns the powerset of a Set.

Usage

powerset(x, simplify = FALSE)

Arguments

Details

A powerset of a set, S, is defined as the set of all subsets of S, including S itself and the empty set.

Value

Set

See Also

Other operators: setcomplement(), setintersect(), setpower(), setproduct(), setsymdiff(), setunion()

Examples

# simplify = FALSE is default
powerset(Set$new(1, 2))
powerset(Set$new(1, 2), simplify = TRUE)

# powerset of intervals
powerset(Interval$new())

# powerset of powersets
powerset(powerset(Reals$new()))
powerset(powerset(Reals$new()))$properties$cardinality

Description

PowersetSet class for symbolic powerset of mathematical sets.

Details

The purpose of this class is to provide a symbolic representation for the powerset of sets that cannot be represented in a simpler class. Whilst this is not an abstract class, it is not recommended to construct this class directly but via the set operation methods.

Super classes

set6::Set -> set6::SetWrapper -> set6::ProductSet -> PowersetSet

Methods

Public methods

• PowersetSet$new()

• PowersetSet$strprint()

• PowersetSet$contains()

• PowersetSet$isSubset()

• PowersetSet$clone()

Method new()

Create a new PowersetSet object. It is not recommended to construct this class directly.

Usage

PowersetSet$new(set)

Arguments

Returns

A new PowersetSet object.

Method strprint()

Creates a printable representation of the object.

Usage

PowersetSet$strprint(n = 2)

Arguments

Returns

A character string representing the object.

Method contains()

Tests if elements x are contained in self.

Usage

PowersetSet$contains(x, all = FALSE, bound = NULL)

Arguments

Returns

If all == TRUE then returns TRUE if all x are contained in self, otherwise FALSE. If all == FALSE returns a vector of logicals corresponding to the length of x, representing if each is contained in self. If bound == TRUE then an element is contained in self if it is on or within the (possibly-open) bounds of self, otherwise TRUE only if the element is within self or the bounds are closed.

Method isSubset()

Tests if x is a (proper) subset of self.

Usage

PowersetSet$isSubset(x, proper = FALSE, all = FALSE)

Arguments

Returns

If all == TRUE then returns TRUE if all x are (proper) subsets of self, otherwise FALSE. If all == FALSE returns a vector of logicals corresponding to the length of x, representing if each is a (proper) subset of self.

Method clone()

The objects of this class are cloneable with this method.

Usage

PowersetSet$clone(deep = FALSE)

Arguments

See Also

Set operations: setunion, setproduct, setpower, setcomplement, setsymdiff, powerset, setintersect

Other wrappers: ComplementSet, ExponentSet, ProductSet, UnionSet

Description

ProductSet class for symbolic product of mathematical sets.

Details

The purpose of this class is to provide a symbolic representation for the product of sets that cannot be represented in a simpler class. Whilst this is not an abstract class, it is not recommended to construct this class directly but via the set operation methods.

Super classes

set6::Set -> set6::SetWrapper -> ProductSet

Active bindings

Methods

Public methods

• ProductSet$new()

• ProductSet$strprint()

• ProductSet$contains()

• ProductSet$clone()

Method new()

Create a new ProductSet object. It is not recommended to construct this class directly.

Usage

ProductSet$new(
  setlist,
  lower = NULL,
  upper = NULL,
  type = NULL,
  cardinality = NULL
)

Arguments

Returns

A new ProductSet object.

Method strprint()

Creates a printable representation of the object.

Usage

ProductSet$strprint(n = 2)

Arguments

Returns

A character string representing the object.

Method contains()

Tests if elements x are contained in self.

Usage

ProductSet$contains(x, all = FALSE, bound = FALSE)

Arguments

Returns

If all == TRUE then returns TRUE if all x are contained in self, otherwise FALSE. If all == FALSE returns a vector of logicals corresponding to the length of x, representing if each is contained in self. If bound == TRUE then an element is contained in self if it is on or within the (possibly-open) bounds of self, otherwise TRUE only if the element is within self or the bounds are closed.

Method clone()

The objects of this class are cloneable with this method.

Usage

ProductSet$clone(deep = FALSE)

Arguments

See Also

Set operations: setunion, setproduct, setpower, setcomplement, setsymdiff, powerset, setintersect

Other wrappers: ComplementSet, ExponentSet, PowersetSet, UnionSet

Description

Used to store the properties of a Set. Though this is not an abstract class, it should never be constructed outside of the Set constructor.

Active bindings

Methods

Public methods

• Properties$new()

• Properties$print()

• Properties$strprint()

• Properties$clone()

Method new()

Creates a new Properties object.

Usage

Properties$new(closure = character(0), cardinality = NULL)

Arguments

Returns

A new Properties object.

Method print()

Prints the Properties list.

Usage

Properties$print()

Returns

Prints Properties list to console.

Method strprint()

Creates a printable representation of the Properties.

Usage

Properties$strprint()

Returns

A list of properties.

Method clone()

The objects of this class are cloneable with this method.

Usage

Properties$clone(deep = FALSE)

Arguments

Description

The mathematical set of rational numbers, defined as the set of numbers that can be written as a fraction of two integers. i.e.

$\\{\frac{p}{q} \ : \ p,q \ \in \ Z, \ q \ne 0 \\}$

where $Z$ is the set of integers.

Details

The ⁠$contains⁠ method does not work for the set of Rationals as it is notoriously difficult/impossible to find an algorithm for determining if any given number is rational or not. Furthermore, computers must truncate all irrational numbers to rational numbers.

Super classes

set6::Set -> set6::Interval -> set6::SpecialSet -> Rationals

Methods

Public methods

• Rationals$new()

• Rationals$contains()

• Rationals$isSubset()

• Rationals$equals()

• Rationals$clone()

Method new()

Create a new Rationals object.

Usage

Rationals$new(lower = -Inf, upper = Inf, type = "()")

Arguments