set6 (version 0.2.4)

FuzzyTuple: Mathematical Fuzzy Tuple

Description

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

Super classes

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

Methods

Public methods

• FuzzyTuple$equals()

• FuzzyTuple$isSubset()

• FuzzyTuple$alphaCut()

• FuzzyTuple$clone()

• set6::Set$add()

• set6::Set$contains()

• set6::Set$multiplicity()

• set6::Set$print()

• set6::Set$remove()

• set6::Set$summary()

• set6::FuzzySet$core()

• set6::FuzzySet$inclusion()

• set6::FuzzySet$initialize()

• set6::FuzzySet$membership()

• set6::FuzzySet$strprint()

• set6::FuzzySet$support()

Method equals()

Tests if two sets are equal.

Usage

FuzzyTuple$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

FuzzyTuple$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	>

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

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 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

deep

Whether to make a deep clone.

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.

See Also

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

Examples

# NOT RUN {
# 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)
# }