Subject Index

  • images ‐level  145
  • images ‐value  145
  • images 165
  • images 193
  • images 195
  • images 195
  • images 196
  • images 200
  • images 165
  • images (bottom)  13, 27
  • cardimages (cardinality)  17
  • images 15
  • images 238, 247
  • cod (codomain)  72
  • images (complement)  17
  • core (core)  21
  • images (Complex numbers)  40
  • dom (domain)  72
  • images (empty fuzzy set)  16
  • images (empty string)  192
  • images (adjunction)  33
  • images (fuzzy powerset)  17
  • images (glb, inf)  12
  • images (height)  15, 76, 96
  • images (image op.)  33
  • images 238, 247
  • images 16, 23
  • images (lub, sup)  13
  • images (norm)  218
  • images 165
  • images (powerser)  13
  • images (preimage op.)  33
  • images (product conjunction)  173
  • images 16
  • images 83
  • images 29
  • *images29
  • images 192
  • images (strong conjunction)  168
  • images 16, 23
  • supp (support)  21
  • images (t‐conorm)  25
  • images (t‐norm)  24
  • images (top)  13, 27
  • images (function space)  33
  • images 72
  • a
  • Absorption  13
  • acceptability degree  134
  • adjoint pair  102, 167
  • adjunction  33
  • adjunction rule  159
  • images ‐cut  19–21, 33, 39, 47, 58, 60, 79, 96, 110, 185, 243
    • strong  19
  • images ‐level, seeimages‐cut
  • alphabet  192
  • alternative hypothesis  145
  • ambiguity  4
  • anti‐fuzzy Lie ideal  232
  • antisymmetry  12
  • apothem  280
  • associativity  13
  • Aumann integral  297
  • automaton
    • finite  191
    • fuzzy  200–205
      • Cartesian composition  204
      • cascade product  201
      • general  205
      • general direct product  200
      • homomorphism  200
      • strong homomorphism  200
      • strong subsystem  204
      • subautomaton  203
      • sum  204
      • wreath product  202
    • multiset
  • b
  • Banach space  219, 237
  • basic logic  169
  • biconditional  157
  • bifurcation  322
  • Bohr cells  286
  • Boolean algebra  35–36, 117
  • Borel set, see set
  • boundary condition  310, 313
  • boundary value problem  310
  • boundedness  304
  • box‐counting algorithm  321
  • c
  • cardinality  17, 21, 210
    • fuzzy  19
    • scalar  17
  • category  7, 260
  • category theory, see theory
  • Cauchy sequence, see sequence
  • central limit theorem  142
  • Chachi‐Taheri‐Viertl algorithm  147
  • chaos game  324
  • chaos theory  320
  • Chu category  100, 264
  • Chu space  100
  • closed ball  236
  • closed form  310
  • commutativity  13, 216
  • compatibility  113, 179
  • compositional inference rule  180–181
  • computation  195, 196
  • computational path  207
  • confidence interval  141, 142
  • confidence length  142
  • confidence level  142, 144
  • confidence level images145
  • confidence limit
  • confidence set  145
  • confidence width  142
  • configuration, see Turing machine
  • conjunction  156
  • connective
    • idempotent  174
    • propositional  157
  • contradictory  158
  • controlling device  193
  • convergence
    • upper measure of  302
  • convex hull  141
  • cover  240, 257, 321
    • refinement  321
  • Cramer's rule  307
  • crisp coefficients  305
  • cumulative distribution function  130
  • d
  • De Morgan's laws  17, 26, 158
  • decision  136
  • decision function  137
  • defuzzification  185
  • degree‐functional  34
  • differential operator  310, 313
  • differentiation
    • sum rule of  300
  • digraph  73
  • discrete probability distribution  130
  • disjunction  156
  • dyadic rationals  255
  • e
  • images ‐ball  236
  • equivalence class  75, 85
  • error
    • false negative  145
    • false positive  145
    • indicator function  146
    • type I 145
    • type II 145
  • estimator
    • bias of  137
    • biased  137
    • consistent  138, 141
    • maximum likelihood  139
    • unbiased  137–138
  • estimator function  137
  • expert system  182
  • exponential, see Heyting algebra
  • expression  194
  • extended addition  297, 300
  • extended solution  310
  • extended subtraction  296
  • extension principle, see principle308
  • f
  • fat fuzzy line, see fuzzy line
  • F‐bounded  261
  • Floyd's algorithm  83
  • forest  90
  • formal language  192
    • fuzzy  196
      • concatenation  197
      • intersection  197
      • Kleene closure  197
      • union  197
  • Fourier series  313
  • Fourier transform  313
  • fractal geometry  320
  • frame  14, 24, 35, 210, 264
  • frequentist approach  141
  • function
    • characteristic  15
    • continuous  303
    • fuzzy continuous  290
    • membership  15, 20
  • function space  33
  • functor  101
  • fundamental fuzzy set  251
  • fundamental set  307
  • fuzzification  184
  • fuzzified partial differential equation  311
  • fuzzifying function  291
    • composition of  291
    • LR type  295
  • fuzzy algorithm  205
  • fuzzy angle  283, 284
  • fuzzy area  278, 282
  • fuzzy Banach space  261
  • fuzzy Bayesian estimation  141
  • fuzzy boundary condition  312
  • fuzzy bunch  291
    • composition of  291
  • fuzzy categories  98–105
  • fuzzy circle  276–280
  • fuzzy circumference  279
  • fuzzy codomain  290
  • fuzzy compactness  257
  • fuzzy connectedness  258
  • fuzzy convergence  301, 302, 304
  • fuzzy coset
  • fuzzy covering  84
  • fuzzy database  107
  • fuzzy diagnosis  108
  • fuzzy distance  269–272, 284
  • fuzzy domain  290, 293
  • fuzzy equality  221, 228
  • fuzzy field  229
  • fuzzy finite state automaton, see automaton
    • subsystem  203
  • fuzzy forest  93
  • fuzzy formal language, see formal language
  • fuzzy function  57, 221, 289–294
    • derivative of  298–300
    • integral of  294–298
    • strong  221
  • fuzzy geometry  269–287
  • fuzzy grammar, see grammar
  • fuzzy graph  89, 90, 97
    • block  93
    • bridge  93
    • cutvertex  93
    • subgraph  91
      • spanning  92
  • fuzzy Hilbert space  263
  • fuzzy hypergraph, see hypergraph
  • fuzzy hypothesis testing  146–148
  • fuzzy ideal  220, 227
  • fuzzy information  143
  • fuzzy information system  143
  • fuzzy initial conditions  305, 309
  • fuzzy interval  41
  • fuzzy lattice  88
  • fuzzy left ideal  220, 227
  • fuzzy Lie ideal  232
  • fuzzy Lie subalgebra  231
  • fuzzy limit
    • of functions  300–304
    • infinite  301
    • of sequences  300–304
    • weak  301
  • fuzzy line
  • fuzzy line segment  281
  • fuzzy linear order  89
  • fuzzy maximum  292
  • fuzzy metric  244, 272
  • fuzzy metric space  240–244
  • fuzzy multiset, see set
  • fuzzy net  256
    • eventually quasi‐coincident  256
    • frequently quasi‐coincident  256
    • quasi‐coincident  256
  • fuzzy norm  230
  • fuzzy normed space  230, 244, 260
    • complete  261
  • fuzzy order relation  87
  • fuzzy ordinary differential equations  304–310
  • fuzzy parameter  140
  • fuzzy partial differential equations  310–313
  • fuzzy partial order  87
  • fuzzy partition  86
  • fuzzy perimeter  282
  • fuzzy plane projective geometry  285
  • fuzzy point  269–272
    • geometry270
    • topology  246
      • images‐cluster  247
  • fuzzy point estimator  140
    • unbiased  140
  • fuzzy polygon
    • images‐sided  281
    • degenerate  281
    • nondegenerate  281
    • regular  280–285
    • strongly nondegenerate  281
  • fuzzy poset  87
  • fuzzy preorder  89
  • fuzzy probability  119
  • fuzzy quantifier  178–179
  • fuzzy random sample  133, 143
  • fuzzy random variable
    • images‐th moment of  134
    • distribution function of  135
    • expectation value of  135
    • expected value of  134
    • identically distributed  136
    • variance of  135
  • fuzzy rectangle  282–284
  • fuzzy relation  71, 109, 179, 221, 291
    • antisymmetric  82
    • binary  75
    • class  84
    • codomain  76
    • compact  106
    • compatibility  84
    • complement  78
    • composition
      • max‐min  79
      • max‐t‐norm  81
      • min‐max  81
    • convex  106
    • cylindrical extension  78
    • domain  76
    • images‐reflexive  82
    • equivalence  83
    • height  76
    • intersection  78
    • irreflexive  82
    • locally reflexive  82
    • membership matrix  76, 79, 81, 109
    • power  83
    • projection  77
    • proximity  84
    • reflexive  82, 86
    • similarity  83
    • symmetric  82
    • tolerance  84
    • transitive  82, 86
      • sup‐min  82
    • transitive closure  82
    • transpose  79
    • union  78
  • fuzzy relational database  107
  • fuzzy restriction  111, 179
  • fuzzy right ideal  220, 227
  • fuzzy ring  227
  • fuzzy separation  250
  • fuzzy set, see set
    • normal  302
  • fuzzy space‐time  285
  • fuzzy sphere  286
  • fuzzy square  284
  • fuzzy statistics  129–153
  • fuzzy subgroup  220, 227
  • fuzzy subgroupoid  220, 227
  • fuzzy subspace  229
  • fuzzy test function  148
  • fuzzy tolology
    • neighborhood
      • Q‐neighborhood  247
  • fuzzy topological space  245–260
    • fuzzy compact  258
    • induced  247
    • interior  247
    • product  248
    • product‐induced  250
    • quasi fuzzy compact  257
    • quasi fuzzy countable compact  257
    • smooth  259
    • subspace  248
    • images251
    • quasi‐images251
    • images251
    • images252
    • images252
    • images254
    • weakly fuzzy compact  257
  • fuzzy topology  245
    • fuzzy set
      • closed  245
      • connected  258
      • images‐closed  247
      • images‐open  247
      • disconnected  258
      • open  245
      • Q‐separated  256, 258
      • quasi‐coincident  246
      • relative closed  248, 258
      • relative open  248
      • separated  256
    • neighborhood
    • relative  248
    • smooth  259
  • fuzzy total order  89
  • fuzzy tree  93
  • fuzzy triangle  283
  • fuzzy trigonometry  284
  • fuzzy vector  106
  • g
  • general fuzzy automaton, see automaton
  • generality  5
  • Gentzen system  159
  • GL‐monoid  103
  • glb  12, 13
  • gld  24
  • Gödel negation  173
  • Gödel logic  174–176
  • Goguen sets, see set
  • grammar  192
    • context‐free  193
    • context‐sensitive  193
    • fuzzy  197
      • context‐free  198
      • context‐sensitive  198
      • equivalent  198
      • recursive  198
      • regular  198
      • unrestricted  198
    • fuzzy multiset
      • regular  213
    • regular  193
    • unrestricted  193
  • graph  89
    • directed  73, 82
  • greatest lower bound  12
  • group  215
    • Abelian  216
    • additive  216
    • element
    • homomorphism  217
    • left coset  216
    • mixed  225
    • multiplicative  216
    • right coset  216
    • subgroup  216
      • normal  216
    • torsion  225
  • groupoid  215
  • h
  • Hausdorff distance  242
  • hedge  183
  • Henstock integral  297
  • Heyting algebra  13, 35
  • Hilbert space  219, 237
  • Hilbert system  159
  • Hukuhara derivative  300
  • hypercomputation  206
  • hypergraph  90, 96, 110
    • fuzzy  96
  • hypothesis testing  144–146
  • i
  • Idempotence  13
  • image operator  33
  • implication
  • imprecision  5
  • incidence matrix  90, 97, 110
  • inf  12
  • initial value problem  305
  • inner product space  219, 237
  • interval arithmetic  47, 55
  • interval condition  308
  • interval estimation  136, 141–142
  • intuitionism  2
  • intuitionistic logic  27, 175–176
  • j
  • join  12
  • l
  • L‐fuzzy normed space  231
  • language, see formal language
    • corecursively enumerable  208
    • recursively enumerable  208
  • lattice  13, 186, 259
  • law
    • contradiction  18, 158
    • excluded middle  18, 158
  • least squares
    • method of  149
  • least upper bound  12
  • Lie algebra  219
  • likelihood function  139
  • lineale  102
  • linear logic  99, 188
  • linear regression  149
  • linear space, see vector space
  • linear transformation  218
  • linearity  297
  • linguistic hedge  54
  • linguistic modifier  7
  • linguistic variable  54–55, 323
  • logic
    • conjunction  11
    • disjunction  11
    • negation  11
  • logically equivalent  157
  • loss function  138
  • lub  12, 13, 24
  • Łukasiewicz logic  165, 171–172
  • m
  • many‐valued logic  161–166
  • maximizing set  291
  • measure  113
    • generalized possibility  115
    • maxitive  114
    • necessity  115
    • possibility  115
  • medical diagnosis  108, 185–187
  • meet  12
  • metric  92, 219, 236, 267
  • metric space  235, 302, 320
    • complete  237
    • diameter  320
  • minimizing set  291
  • modus ponens  159, 172, 188
    • compositional  181
  • monoid  215
  • multiset, see set
  • n
  • natural frequency  309
  • negation  156
  • neoclassical analysis  300
  • neutrosophic set, see set
  • noncommutative geometry  286
  • nondegenerate fuzzy polygon, see fuzzy polygon
  • nonfuzzy domain  293
  • nonfuzzy function  293
  • nonfuzzy interval  294
  • nonstandard analysis  29
  • norm
    • naturally defined  219, 237
  • normal fuzzy random sample  136
  • normal set representation  135
  • null hypothesis  145
  • number
    • fuzzy  39–68, 106, 243, 270
      • arithmetic  46–51
      • arithmetic  244–244
      • comparison  51–53
      • complex  40, 59
      • discrete  40, 49
      • exponential  44
      • Gaussian  42
      • generalized  46, 49–51, 187
      • L‐R  44, 296
      • lower semi‐continuous  243
      • negative  39
      • positive  39
      • quadratic  43
      • quasi‐Gaussian  42, 66
      • trapezoidal  41, 108
      • triangular  40, 48, 57, 151, 274, 275, 310
      • upper semi‐continuous  242
    • hyperreal  29
    • real
      • as fuzzy  40
  • p
  • P system  212
  • parallelism  276
  • partial differential equation
    • nonhomogeneous  312
  • partition  75
  • picture fuzzy set, see set
  • point estimation  136–140
  • point‐slope form  273
  • poset  12
    • monoidal  102
  • possibility theory  115–118
  • predistor variable  148
  • preimage operator  33, 99
  • principle
  • probability density function  131
    • conditional  132
  • probability mass function  130
    • conditional  132
  • probability theory, see theory
  • product logic  172–174
  • propositional logic  158
  • proximity relation  84
  • r
  • random polygons  285
  • random set  133
  • random variable
    • images‐th moment of  131
    • distribution function of  130
    • expectation value of  131
    • mean value of  131
    • standard deviation of  131
    • variance of  131
  • randomness  5
  • rational pavelka logic  172
  • recursive language  193
  • reflexivity  12
  • regression equation  148
  • regular fuzzy circle, see fuzzy circle
  • regular fuzzy polygon, see fuzzy polygon
  • relation
    • antisymmetric  72
    • binary  12, 71
    • codomain  72
    • composition  73
    • difference  72
    • domain  72
    • equivalence  75, 83, 85, 110
    • intersection  72
    • power  73
    • reflexive  72
    • step  196
    • symmetric  72
    • transitive  72, 74
    • transitive closure  74
    • union  72
  • relatively computable function  194
  • resolution form  79, 87
  • resonance  309
  • resonant frequency  309
  • response variable  148
  • Riemann integral
    • generalization of  296
  • rigid translations  282
  • ring  217
  • risk function  138
  • s
  • sagittal diagram  76
  • sample fuzzy information  143
  • semigroup  215
  • semiorder  256
  • sequence  236
  • set  7
    • Borel  241
    • crisp  16
    • directed  256
    • fuzzy  6, 7
      • “intuitionistic”  27, 187, 210
      • “intuitionistic”‐L  28
      • algebraic product  17, 225
      • bounded  21
      • Cartesian product  77, 106
      • complement  17
      • constant  245
      • convex  18, 39, 243
      • core  21, 39
      • height  15
      • intersection  17
      • interval valued  22
      • L  26, 28, 210, 231
      • multiset  9, 209
      • normal  15, 243, 302
      • picture  30, 38
      • powerset  17
      • quasi‐coincident  256
      • subnormal  15
      • subset  17
      • support  21
      • type images22
      • type 2, 22
      • union  17
    • Goguen  26, 99
    • hybrid
      • L‐fuzzy  210
    • L‐multi‐fuzzy  210
    • multi‐fuzzy  210
    • multiset  9
    • neutrosophic  29–30
    • powerset  13, 31, 158
    • rough  7, 8, 34
    • sharp  16
    • soft  31, 237
      • picture fuzzy  32
    • universe  15
  • sigma‐algebra (images‐algebra)  133, 241
  • sigma‐ring (images‐ring)  241
  • significance level  142
  • significance level images145
  • similarity class  85
  • simplification rule  159
    • generalized  182
  • smooth fuzzy topology
    • see fuzzy topology  259
  • smooth topological space  259, 268
  • soft set, see set
  • solution
    • general  307
    • particular  307
  • sorites paradox  1, 4, 8
  • statement form  157, 158, 169
  • statistical regression  148–150
  • step, see relation
  • strongly nondegenerate fuzzy polygon, see fuzzy polygon281
  • subcover  240, 257, 321
  • sup  12
  • syllogism  155
  • u
  • uncertainty  5
  • unit interval  24, 29
  • unit laws  13
  • Urysohn's Lemma  240, 254
  • v
  • vague binary operation  221, 228
  • vague group  221, 222, 227
  • vague homomorphism  224, 233
  • vague ideal  229
  • vague monoid  222
  • vague ring  228
  • vague semigroup  222
    • Abelian  222
  • vague subgroup  224
  • vagueness  1–9
    • view
      • epistemic  2
      • ontic  2, 6
      • semantic  2, 7, 14
  • vector space  218, 237
    • inner product  219
    • normed  219
  • vibrating system  309
  • w
  • well‐posed problem  310
  • Wronski determinant  307
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