614:
430:
516:
316:
247:
122:
49:
307:
278:
203:
174:
142:
96:
76:
432:
where the down arrow means that the term on the left side of it is defined. Then it becomes possible to define the strong equality in the following way:
55:, that states that on a given argument either both functions are undefined, or both are defined and their values on that arguments are equal.
435:
595:
655:
684:
679:
425:{\displaystyle (y_{1}\sim y_{2}):\Leftrightarrow ((y_{1}\downarrow \lor y_{2}\downarrow )\longrightarrow y_{1}=y_{2}),}
674:
648:
641:
208:
101:
591:
52:
625:
34:
585:
283:
254:
179:
150:
127:
81:
61:
668:
621:
533:
20:
613:
511:{\displaystyle (f\simeq g):\Leftrightarrow (\forall x.(f(x)\sim g(x))).}
564:
313:
Some authors are using "quasi-equality", which is defined like this:
587:
Computability, an introduction to recursive function theory
569:
629:
438:
319:
286:
257:
211:
182:
153:
130:
104:
84:
64:
37:
510:
424:
301:
272:
241:
197:
168:
136:
116:
90:
70:
43:
565:"A Set Theory with Support for Partial Functions"
563:Farmer, William M.; Guttman, Joshua D. (2000).
649:
8:
590:. Cambridge University Press. p. 251.
656:
642:
58:For example, if we have partial functions
437:
410:
397:
378:
362:
340:
327:
318:
285:
256:
210:
181:
152:
129:
103:
83:
63:
36:
550:
525:
16:Equality operator on partial functions
7:
610:
608:
628:. You can help Knowledge (XXG) by
460:
14:
612:
502:
499:
496:
490:
481:
475:
469:
457:
451:
439:
416:
390:
387:
384:
368:
355:
352:
346:
320:
296:
290:
267:
261:
236:
230:
221:
215:
192:
186:
163:
157:
1:
51:) is an equality operator on
701:
607:
685:Equivalence (mathematics)
534:"Kleene equality in nLab"
242:{\displaystyle f(x)=g(x)}
117:{\displaystyle f\simeq g}
584:Cutland, Nigel (1980).
44:{\displaystyle \simeq }
624:-related article is a
512:
426:
303:
274:
243:
199:
170:
138:
118:
92:
72:
45:
513:
427:
304:
275:
244:
205:are both defined and
200:
171:
139:
124:means that for every
119:
93:
73:
46:
680:Computability theory
436:
317:
302:{\displaystyle g(x)}
284:
273:{\displaystyle f(x)}
255:
209:
198:{\displaystyle g(x)}
180:
169:{\displaystyle f(x)}
151:
128:
102:
82:
62:
35:
309:are both undefined.
508:
422:
299:
270:
239:
195:
166:
134:
114:
88:
68:
41:
675:Mathematics stubs
637:
636:
597:978-0-521-29465-2
137:{\displaystyle x}
91:{\displaystyle g}
71:{\displaystyle f}
53:partial functions
692:
658:
651:
644:
616:
609:
601:
577:
576:
560:
554:
548:
542:
541:
530:
517:
515:
514:
509:
431:
429:
428:
423:
415:
414:
402:
401:
383:
382:
367:
366:
345:
344:
332:
331:
308:
306:
305:
300:
279:
277:
276:
271:
248:
246:
245:
240:
204:
202:
201:
196:
175:
173:
172:
167:
143:
141:
140:
135:
123:
121:
120:
115:
97:
95:
94:
89:
77:
75:
74:
69:
50:
48:
47:
42:
700:
699:
695:
694:
693:
691:
690:
689:
665:
664:
663:
662:
605:
598:
583:
580:
562:
561:
557:
549:
545:
532:
531:
527:
523:
434:
433:
406:
393:
374:
358:
336:
323:
315:
314:
282:
281:
253:
252:
207:
206:
178:
177:
149:
148:
126:
125:
100:
99:
80:
79:
60:
59:
33:
32:
29:strong equality
25:Kleene equality
17:
12:
11:
5:
698:
696:
688:
687:
682:
677:
667:
666:
661:
660:
653:
646:
638:
635:
634:
617:
603:
602:
596:
579:
578:
555:
543:
524:
522:
519:
507:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
421:
418:
413:
409:
405:
400:
396:
392:
389:
386:
381:
377:
373:
370:
365:
361:
357:
354:
351:
348:
343:
339:
335:
330:
326:
322:
311:
310:
298:
295:
292:
289:
269:
266:
263:
260:
249:
238:
235:
232:
229:
226:
223:
220:
217:
214:
194:
191:
188:
185:
165:
162:
159:
156:
133:
113:
110:
107:
87:
67:
40:
15:
13:
10:
9:
6:
4:
3:
2:
697:
686:
683:
681:
678:
676:
673:
672:
670:
659:
654:
652:
647:
645:
640:
639:
633:
631:
627:
623:
618:
615:
611:
606:
599:
593:
589:
588:
582:
581:
574:
570:
566:
559:
556:
552:
547:
544:
539:
535:
529:
526:
520:
518:
505:
493:
487:
484:
478:
472:
466:
463:
454:
448:
445:
442:
419:
411:
407:
403:
398:
394:
379:
375:
371:
363:
359:
349:
341:
337:
333:
328:
324:
293:
287:
264:
258:
250:
233:
227:
224:
218:
212:
189:
183:
160:
154:
147:
146:
145:
131:
111:
108:
105:
85:
65:
56:
54:
38:
30:
26:
22:
630:expanding it
619:
604:
586:
572:
568:
558:
553:, p. 3.
551:Cutland 1980
546:
537:
528:
312:
57:
28:
24:
18:
622:mathematics
575:(1): 59–78.
538:ncatlab.org
21:mathematics
669:Categories
521:References
455::⇔
350::⇔
485:∼
461:∀
446:≃
391:⟶
385:↓
372:∨
369:↓
334:∼
109:≃
39:≃
594:
620:This
27:, or
626:stub
592:ISBN
280:and
176:and
78:and
251:or
31:, (
19:In
671::
573:66
571:.
567:.
536:.
144::
98:,
23:,
657:e
650:t
643:v
632:.
600:.
540:.
506:.
503:)
500:)
497:)
494:x
491:(
488:g
482:)
479:x
476:(
473:f
470:(
467:.
464:x
458:(
452:)
449:g
443:f
440:(
420:,
417:)
412:2
408:y
404:=
399:1
395:y
388:)
380:2
376:y
364:1
360:y
356:(
353:(
347:)
342:2
338:y
329:1
325:y
321:(
297:)
294:x
291:(
288:g
268:)
265:x
262:(
259:f
237:)
234:x
231:(
228:g
225:=
222:)
219:x
216:(
213:f
193:)
190:x
187:(
184:g
164:)
161:x
158:(
155:f
132:x
112:g
106:f
86:g
66:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.