Knowledge (XXG)

Special case

Source 📝

345: 238: 461: 393: 283: 485: 78:
is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. If
160: 241: 292: 526: 75: 286: 177: 194: 117: 409: 129: 98:
is a special case which is in some way qualitatively different from almost all of the cases allowed.
354: 359: 156: 396: 259: 114:(but not all rectangles are squares); therefore the square is a special case of the rectangle. 506: 470: 95: 67: 520: 17: 400: 35: 507:
Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures
111: 253: 185: 27:
Specific, usually well-known application of a mathematical rule or law
31: 510:. United Kingdom, Taylor & Francis, 2005. 27. 473: 412: 362: 295: 262: 197: 136:
has no primitive solutions in positive integers with
479: 455: 387: 340:{\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}} 339: 277: 232: 94:can also be immediately deduced to be false. A 148:all greater than 2, specifically, the case of 106:Special case examples include the following: 8: 124:has no solutions in positive integers with 472: 417: 411: 367: 361: 321: 300: 294: 261: 214: 202: 196: 82:is true, one can immediately deduce that 233:{\displaystyle a^{p}\equiv a{\pmod {p}}} 62:but not vice versa, or equivalently, if 497: 456:{\displaystyle e^{ix}=\cos x+i\sin x} 7: 329: 222: 25: 184:is a prime number, then for any 504:Brown, James Robert.  322: 215: 54:precisely if every instance of 333: 323: 310: 304: 272: 266: 226: 216: 161:generalized Riemann hypothesis 1: 388:{\displaystyle e^{i\pi }=-1} 34:, especially as applied in 543: 159:is a special case of the 287:Euler's totient function 278:{\displaystyle \phi (n)} 86:is true as well, and if 256:positive integers, and 240:" is a special case of 178:Fermat's little theorem 128:, is a special case of 58:is also an instance of 481: 457: 399:which states "for any 389: 341: 279: 234: 482: 458: 395:is a special case of 390: 342: 280: 235: 118:Fermat's Last Theorem 480:{\displaystyle \pi } 471: 463:", in the case that 410: 360: 347:", in the case that 293: 260: 195: 244:, which states "if 180:, which states "if 163:, in the case that 527:Mathematical logic 477: 453: 385: 351:is a prime number. 337: 275: 230: 157:Riemann hypothesis 130:Beal's conjecture 18:Particularization 16:(Redirected from 534: 511: 502: 486: 484: 483: 478: 466: 462: 460: 459: 454: 425: 424: 394: 392: 391: 386: 375: 374: 355:Euler's identity 350: 346: 344: 343: 338: 336: 314: 313: 284: 282: 281: 276: 239: 237: 236: 231: 229: 207: 206: 183: 151: 147: 143: 139: 135: 127: 123: 110:All squares are 93: 89: 85: 81: 73: 65: 61: 57: 53: 41: 21: 542: 541: 537: 536: 535: 533: 532: 531: 517: 516: 515: 514: 503: 499: 494: 469: 468: 464: 413: 408: 407: 397:Euler's formula 363: 358: 357: 348: 296: 291: 290: 258: 257: 242:Euler's theorem 198: 193: 192: 181: 149: 145: 141: 137: 133: 125: 121: 104: 96:degenerate case 91: 87: 83: 79: 71: 63: 59: 55: 51: 39: 28: 23: 22: 15: 12: 11: 5: 540: 538: 530: 529: 519: 518: 513: 512: 496: 495: 493: 490: 489: 488: 476: 452: 449: 446: 443: 440: 437: 434: 431: 428: 423: 420: 416: 384: 381: 378: 373: 370: 366: 352: 335: 332: 328: 325: 320: 317: 312: 309: 306: 303: 299: 274: 271: 268: 265: 228: 225: 221: 218: 213: 210: 205: 201: 175: 171:) = 1 for all 153: 115: 103: 100: 68:generalization 48:specialization 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 539: 528: 525: 524: 522: 509: 508: 501: 498: 491: 474: 450: 447: 444: 441: 438: 435: 432: 429: 426: 421: 418: 414: 405: 402: 398: 382: 379: 376: 371: 368: 364: 356: 353: 330: 326: 318: 315: 307: 301: 297: 288: 269: 263: 255: 251: 247: 243: 223: 219: 211: 208: 203: 199: 190: 187: 179: 176: 174: 170: 166: 162: 158: 155:The unproven 154: 131: 119: 116: 113: 109: 108: 107: 101: 99: 97: 77: 76:limiting case 69: 49: 45: 37: 33: 19: 505: 500: 403: 249: 245: 188: 172: 168: 164: 105: 47: 44:special case 43: 29: 401:real number 50:of concept 36:mathematics 492:References 112:rectangles 90:is false, 38:, concept 475:π 448:⁡ 433:⁡ 380:− 372:π 316:≡ 302:φ 264:ϕ 209:≡ 150:x = y = z 134:a + b = c 122:a + b = c 521:Category 126:n > 2 102:Examples 289:, then 254:coprime 191:, then 186:integer 132:, that 120:, that 144:, and 66:is a 42:is a 32:logic 252:are 248:and 74:. A 445:sin 430:cos 327:mod 285:is 220:mod 70:of 46:or 30:In 523:: 467:= 406:: 173:n. 140:, 487:. 465:x 451:x 442:i 439:+ 436:x 427:= 422:x 419:i 415:e 404:x 383:1 377:= 369:i 365:e 349:n 334:) 331:n 324:( 319:1 311:) 308:n 305:( 298:a 273:) 270:n 267:( 250:a 246:n 227:) 224:p 217:( 212:a 204:p 200:a 189:a 182:p 169:n 167:( 165:χ 152:. 146:z 142:y 138:x 92:A 88:B 84:A 80:B 72:A 64:B 60:B 56:A 52:B 40:A 20:)

Index

Particularization
logic
mathematics
generalization
limiting case
degenerate case
rectangles
Fermat's Last Theorem
Beal's conjecture
Riemann hypothesis
generalized Riemann hypothesis
Fermat's little theorem
integer
Euler's theorem
coprime
Euler's totient function
Euler's identity
Euler's formula
real number
Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures
Category
Mathematical logic

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.