- ‐level 145
- ‐value 145
- 165
- 193
- 195
- 195
- 196
- 200
- 165
- (bottom) 13, 27
- card (cardinality) 17
- 15
- 238, 247
- cod (codomain) 72
- (complement) 17
- core (core) 21
- (Complex numbers) 40
- dom (domain) 72
- (empty fuzzy set) 16
- (empty string) 192
- (adjunction) 33
- (fuzzy powerset) 17
- (glb, inf) 12
- (height) 15, 76, 96
- (image op.) 33
- 238, 247
- 16, 23
- (lub, sup) 13
- (norm) 218
- 165
- (powerser) 13
- (preimage op.) 33
- (product conjunction) 173
- 16
- 83
- 29
- * 29
- 192
- (strong conjunction) 168
- 16, 23
- supp (support) 21
- (t‐conorm) 25
- (t‐norm) 24
- (top) 13, 27
- (function space) 33
- 72
- a
- Absorption 13
- acceptability degree 134
- adjoint pair 102, 167
- adjunction 33
- adjunction rule 159
- ‐cut 19–21, 33, 39, 47, 58, 60, 79, 96, 110, 185, 243
- ‐level, see‐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
- 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 145
- 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
- convex hull 141
- cover 240, 257, 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
- digraph 73
- discrete probability distribution 130
- disjunction 156
- dyadic rationals 255
- f
- fat fuzzy line, see fuzzy line
- F‐bounded 261
- Floyd's algorithm 83
- forest 90
- formal language 192
- 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
- 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
- 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
- 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
- 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
- fuzzy point estimator 140
- fuzzy polygon
- ‐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
- ‐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
- ‐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
- 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
- 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
- fuzzy topological space 245–260
- fuzzy topology 245
- fuzzy set
- 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 193
- unrestricted 193
- graph 89
- greatest lower bound 12
- group 215
- 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
- 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
- l
- L‐fuzzy normed space 231
- language, see formal language
- corecursively enumerable 208
- recursively enumerable 208
- lattice 13, 186, 259
- law
- least squares
- 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
- 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
- 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
- p
- P system 212
- parallelism 276
- partial differential equation
- partition 75
- picture fuzzy set, see set
- point estimation 136–140
- point‐slope form 273
- poset 12
- possibility theory 115–118
- predistor variable 148
- preimage operator 33, 99
- principle
- probability density function 131
- probability mass function 130
- probability theory, see theory
- product logic 172–174
- propositional logic 158
- proximity relation 84
- q
- quantifier
- quantum geometry 285
- r
- random polygons 285
- random set 133
- random variable
- ‐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
- 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 22
- type 2, 22
- union 17
- Goguen 26, 99
- hybrid
- 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
- universe 15
- sigma‐algebra (‐algebra) 133, 241
- sigma‐ring (‐ring) 241
- significance level 142
- significance level 145
- similarity class 85
- simplification rule 159
- smooth fuzzy topology
- smooth topological space 259, 268
- soft set, see set
- solution
- sorites paradox 1, 4, 8
- statement form 157, 158, 169
- statistical regression 148–150
- step, see relation
- strongly nondegenerate fuzzy polygon, see fuzzy polygon 281
- subcover 240, 257, 321
- sup 12
- syllogism 155
- t
- t‐conorm, see triangular conorm
- t‐norm, see triangular norm
- tautology 158
- test function 145
- theoretical physics 285–287
- theory
- category 7, 98–105
- probability 5
- thick fuzzy circle, see fuzzy circle
- thin fuzzy line, see fuzzy line
- three‐valued logic 162
- topological space 237
- topology 237
- transition function, see Turing machine 195
- transition rule 191
- transitivity 12
- tree 90
- triangular conorm 24, 25, 78
- triangular norm 24, 77, 78, 81, 82, 117, 167, 230, 241
- Turing machine
- computation 196
- configuration 196–196
- fuzzy
- deterministic 206
- nondeterministic 206
- step 207
- nondeterministic 208
- oracle 194
- transition function 195
- two‐point form 273
- type 7
- 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
- vague subgroup 224
- vagueness 1–9
- view
- epistemic 2
- ontic 2, 6
- semantic 2, 7, 14
- vector space 218, 237
- vibrating system 309
- w
- well‐posed problem 310
- Wronski determinant 307
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