FuzzyMultiset: Mathematical Fuzzy Multiset

Description Details Super classes Methods See Also Examples

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

Inherited methods

Method equals()

Tests if two sets are equal.

Usage
FuzzyMultiset$equals(x, all = FALSE)
Arguments
x

Set or vector of Sets.

all

logical. If FALSE tests each x separately. Otherwise returns TRUE only if all x pass test.

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:

Equal ==
Not equal !=
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
x

any. Object or vector of objects to test.

proper

logical. If TRUE tests for proper subsets.

all

logical. If FALSE tests each x separately. Otherwise returns TRUE only if all x pass test.

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:

Subset <
Proper Subset <=
Superset >
Proper Superset >=
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_α = {x ε F | m ≥ α}

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
alpha

numeric in [0, 1] to determine which elements to return

strong

logical, if FALSE (default) then includes elements greater than or equal to alpha, otherwise only strictly greater than

create

logical, if FALSE (default) returns the elements in the alpha cut, otherwise returns a crisp set of the elements

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
deep

Whether to make a deep clone.

See Also

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

Examples

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# 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)

set6 documentation built on Oct. 18, 2021, 5:06 p.m.