Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory

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k-Monotone measures. An alternative name for Choquet capacities of order

Linguistic variable. A variable whose states are fuzzy intervals assigned to

relevant linguistic terms.

Lower probability function. For any given set D of probability distribution

functions p on a ﬁnite set X, the lower probability function

m is deﬁned

for all sets A 債 X by the formula

Measure of fuzziness of a fuzzy set. A functional that usually measures the

lack of distinction between a given fuzzy set and its complement.

Möbius representation. Given a monotone measure m on ·X, P(X)Ò, where X

is ﬁnite, its Möbius representation is a set function m deﬁned for all

A ŒP(X) by the formula

Monotone measure. A set function m :C Æ ⺢

deﬁned on a nonempty family

C of a given set X for which m(Ø) = 0 and m(A) £ m(B) for all A, B Œ C

such that A 債 B.

Necessity measure. A superadditive measure m for which m(A « B) =

min{m(A), m(B)}.

Nested family of crisp sets. Family of sets {A

, A

,...,A

} such that A

債 A

i+1

for all i = 1,2,...,n - 1.

Nonspeciﬁcity. The type of information-based uncertainty that is measured

by the Hartley measure and its various generalizations, or, alternatively, by

the difference between the maximum and minimum values of the Shannon

entropy within a given convex set of probability distributions.

Partial ordering. A binary relation on X

that is reﬂexive, antisymmetric, and

transitive.

Partition of X. A disjoint family {A

, A

,...A

} of nonempty subsets of X such

that .

Plausibility measure. Alternating Choquet capacity of inﬁnite order.

Possibility measure. A subadditive measure m for which m(A » B) =

max{m(A), m(B)}.

Possibility proﬁle. A tuple consisting of all values of a basic possibility func-

tion in decreasing order.

Relation. A subset (crisp or fuzzy) of a Cartesian product.

mA B

BB A

()

() ()

1 m .

m Apx

()

inf .

452 APPENDIX F. GLOSSARY OF KEY CONCEPTS

Shannon cross-entropy. The functional

where p and q are probability distribution functions and X is a ﬁnite set.

When X = ⺢, p and q are probability density functions, and

Shannon entropy. The functional , where

p is a probability distribution function on a ﬁnite set X.This functional mea-

sures the amount of uncertainty associated with function p.

Standard complement of fuzzy set A. Fuzzy set whose membership function

is deﬁned by 1 - A(x) for all x ŒX.

Standard intersection of fuzzy sets A and B. Fuzzy set whose membership

function is deﬁned by min{A(x), B(x)} for all x ŒX.

Standard union of fuzzy sets A and B. Fuzzy set whose membership function

is deﬁned by max {A(x), B(x)} for all x ŒX.

Strong a-cut of fuzzy set A. Crisp set

A = {x|A(x) ≥ a}.

a-cut of fuzzy set A. Crisp set

A = {x|A(x) ≥ a}.

Subadditive measure. A monotone measure m on ·X, CÒ for which m(A » B)

£ m(A) + m(B) whenever A, B, A » B Œ C.

Sugeno l-measure. A monotone measure

m on ·X, P(X)Ò such that

for any disjoint subsets A, B of X, where l Œ (-1, •).

Superadditive function. A monotone measure on ·X, CÒ for which m(A » B)

≥ m(A) + m(B) whenever A, B, A » B Œ C.

Triangular conorm (t-conorm). A function u :[0, 1]

Æ [0, 1] that is commu-

tative, associative, monotone nondecreasing, and such that u(a, 0) = a for

all a Œ [0, 1]. Triangular conorms qualify as union operations of fuzzy sets.

Triangular norm (t-norm). A function i :[0, 1]

Æ [0, 1] that is commutative,

associative, monotone nondecreasing, and such that i(a, 1) = a for all a Œ

[0, 1]. Triangular norms qualify as intersection operations of fuzzy sets.

Uncertainty-based information. The difference between a priori uncertainty

and a reduced a posteriori uncertainty.

Universal set. The collection of all objects that are of interest in a given

application.

llll

mmmlmmAB A B A B»

()

()()

Spx x X px px

()

[]

() ()

log

Spx qc x px

() ()

[]

()

, log .⺢

⺢

Spx qc x X px

() ()

[]

()

, log ,

APPENDIX F. GLOSSARY OF KEY CONCEPTS 453

Upper probability function. For any given set D of probability distribution

functions p on a ﬁnite set X, upper probability function

is deﬁned for all

A 債 X by the formula

m Apx

()

sup .

454 APPENDIX F. GLOSSARY OF KEY CONCEPTS

APPENDIX G

GLOSSARY OF SYMBOLS

455

GENERAL SYMBOLS

{x, y, ...} Set of elements x, y, ...

{x | p(x)} Set determined by property p

·x

, x

,...,x

Ò n tuple

] Matrix

, x

,...,x

] Vector

[a, b] Closed interval of real numbers between a and b

[a, b), (b, a] Interval of real numbers closed in a and open in b

(a, b) Open interval of real numbers

[a, •] Set of real numbers greater than or equal to a

A, B, C, ... Arbitrary sets (crisp or fuzzy)

X Universal set (universe of discourse)

Ø Empty set

x ŒA Element x belongs to crisp set A

Characteristic function of crisp set A

A (x) Membership grade of x in fuzzy set A

A = B Equality of sets (crisp or fuzzy)

A π B Inequality of sets (crisp or fuzzy)

A - B Difference of crisp sets (A - B = {x | x Œ A and x 僆 A})

A 債 B Set inclusion of crisp or fuzzy sets

A Ã B Proper set inclusion (A 債 B and A π B) of crisp or fuzzy

sets

SUB(A, B) Degree of subsethood of A in B

P(X) Set of all crisp subsets of X (power set)

F(X) Set of all fuzzy subsets of X (fuzzy power set)

|A| Cardinality of crisp or fuzzy set A (sigma count)

Height of fuzzy set A

Uncertainty and Information: Foundations of Generalized Information Theory, by George J. Klir

Complement of crisp set A

A « B Intersection of crisp sets

A » B Union of crisp sets

A ¥ B Cartesian product of crisp sets A and B

Cartesian product A ¥ A

[a, b]

Cartesian product [a, b] ¥ [a, b]

X Æ Y Function from X to Y

R(X, Y) Relation on X ¥ Y (crisp or fuzzy)

R ° Q Standard composition of relations R and Q (crisp or

fuzzy)

Q Standard join of relations R and Q (crisp or fuzzy)

-1

Inverse of a binary relation (crisp or fuzzy)

< Less than

£ Less than or equal to (also used for a partial ordering)

x Ÿ y Meet (greatest lower bound) of x and y in a lattice or

logic conjunction

x ⁄ y Join (least upper bound) of x and y in a lattice or logic

disjunction

x|yxgiven y

x ﬁ yximplies y

x ¤ yxif and only if y

" For all (universal quantiﬁer)

$ There exists (existential quantiﬁer)

S Summation

P Product

max{a

, a

,...,a

} Maximum of {a

, a

,...,a

}

min{a

, a

,...,a

} Minimum of {a

, a

,...,a

}

i, j, k Arbitrary identiﬁers (indices)

I, J, K General sets of identiﬁers

⺞ Set of positive integers (natural numbers)

⺞

Set {1, 2,...,n}

⺢ Set of all real numbers

⺢

Set of nonnegative real numbers

⺪ Set of all integers

p(A) Partition of crisp set A

SPECIAL SYMBOLS

AU Aggregate uncertainty in Dempster–Shafer theory

A a-Cut of fuzzy set A for some value a Œ [0, 1]

A Strong a-cut of fuzzy set A for some value a Œ [0, 1]

Bel Belief measure

c Fuzzy complement

456 APPENDIX G. GLOSSARY OF SYMBOLS

(C)兰

fdm Choquet integral of function f with respect to monotone

measure m

d(A) Defuzziﬁed value of fuzzy set A

D Convex set of probability distributions (credal set)

F Set of focal elements in evidence theory

·F, mÒ Body of evidence

i Fuzzy intersection or t-norm

m Möbius representation of a lower probability function

Nec Necessity measure

Nec

Necessity measure associated with a fuzzy proposition “V

is F”

Pl Plausibility measure

Pos Possibility measure

Pos

Possibility measure associated with a fuzzy proposition

“V is F”

, p

Marginal probability distributions

r Possibility proﬁle

u Fuzzy union or t-conorm

f(A) Measure of fuzziness of fuzzy set A

GH Generalized Hartley measure

GS Generalized Shannon entropy

h Averaging operation

H Hartley measure

HL Hartley-like measure (for convex subsets of ⺢

)

S Shannon entropy

S Minimal value of Shannon entropy within a given convex

set of probability distributions (a generalized Shannon

entropy)

Aggregate uncertainty for convex sets of probability

distributions

T Information transmission

TU Disaggregated total uncertainty TU =·GH, S

- GHÒ

TU Alternative disaggregated total uncertainty

TU =·S

- S,

SÒ

UU-Uncertainty (measure of nonspeciﬁcity in the theory

of graded possibilities)

m Monotone measure

m Sugeno l-measure

Lower probability function

Upper probability function

APPENDIX G. GLOSSARY OF SYMBOLS 457

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