84:
74:
53:
1001:. Yes, I can see how this is horribly non-constructive. To build E, I need to somehow pick uncountably many real numbers, and do so in such a way that no two of them differ by a dyadic fraction. Ouch. Anyway, I think that this is what the above paragraph was trying to telegraph. It took me a while to figure this out. That paragraph should be expanded into something less concise, and easier to understand. A reference would be nice, too.
22:
1059:. This too should be mentioned in the article. Anyway, this is all "original research" on my part, but it seems obvious in retrospect, and surely there is some publication that actually spells all of this out in detail.
690:
140:
517:
975:
is unambiguous (because x and y differ in only finitely many places, and there are only countably many different y grand total.) And this can be done for any other equivalence class
457:
270:
231:
609:
1057:
1035:
720:
631:
805:
547:
348:
322:
302:
999:
944:
918:
883:
857:
831:
973:
769:
749:
408:
388:
368:
696:. This cleans up four things: first, it avoids having to blather too much about the equivalence relation. Second, it allows one to make statements about the
722:
is the set of all finite-length binary strings, it allows the construction of the quotient space to resemble that of the construction of open sets in the
1087:
130:
172:
back in 1969, or at least in the controversy surrounding the book. I'd love to write a bit more about this, but I'm not too familiar with the subject.
1082:
106:
1060:
97:
58:
1009:
726:: specifically, by identifying all of the open sets in Baire space. Or perhaps, better yet, the same construction, but on the
777:
for binary strings) and the tree basis topology for the same (which only uses prefixes of finite-length binary strings).
33:
640:
166:, and recall that the inability of linear models to learn XOR (parity of two inputs) was a major point in the book
469:
723:
390:
differ at finite number of coordinates. Having such representatives, we can map all of them to 0; the rest of
21:
1064:
417:
39:
83:
236:
207:
833:
contains only a countable number of representatives (because m and n are countable). Next, for each
552:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1040:
1018:
703:
614:
181:
168:
162:
Parity functions have received some interest in (theoretical) machine learning as well; see e.g.
89:
783:
73:
52:
526:
414:
The first sentence is fine- we have lots of parity functions f. The second sentence introduces
327:
697:
307:
275:
978:
923:
888:
862:
836:
810:
1013:
634:
201:
949:
807:
such equivalence classes; the cardinality of E is the continuum. Each equivalence class
693:
754:
734:
393:
373:
353:
1076:
173:
774:
727:
464:
460:
102:
1005:
611:
for some integers m,n. OK, so far. It would be a lot cleaner to just say that
79:
463:. The second sentence then seems to be saying that one should construct the
304:. It is enough to take one representative per equivalence class of relation
1008:, or rather, it's homeomorphic to it. The homeomorphism is provided by the
520:
1068:
186:
204:
it can be easily proved that parity functions exist and there are
549:
where if x,y are the binary expansions of real numbers x,y with
637:, aka the set of all finite-length binary strings. Then define
523:
string idea. That's fine, too. Intuitively, it is saying that
15:
1004:
Oh, duhh! The collection of equivalence classes is just the
196:
I'm having trouble making sense of the following paragraph:
233:
many of them - as many as the number of all functions from
163:
1012:, which provides a one-to-one and onto mapping between
1043:
1021:
981:
952:
926:
891:
865:
839:
813:
786:
757:
737:
706:
643:
617:
555:
529:
472:
420:
396:
376:
356:
330:
310:
278:
239:
210:
192:
equivalence classes of the infinite parity function?
101:, a collaborative effort to improve the coverage of
1051:
1029:
993:
967:
938:
912:
877:
851:
825:
799:
763:
743:
714:
684:
625:
603:
541:
511:
451:
402:
382:
362:
342:
316:
296:
264:
225:
730:. Fourth, it cleans up the relationship between
685:{\displaystyle E=\{0,1\}^{\omega }/\mathbb {D} }
459:without naming it. But it has a name, its the
8:
663:
650:
512:{\displaystyle E=\{0,1\}^{\omega }/\approx }
492:
479:
440:
427:
291:
279:
253:
240:
19:
47:
1045:
1044:
1042:
1023:
1022:
1020:
980:
951:
925:
890:
864:
838:
812:
791:
785:
756:
736:
708:
707:
705:
678:
677:
672:
666:
642:
619:
618:
616:
595:
586:
554:
528:
501:
495:
471:
443:
419:
395:
375:
355:
329:
309:
277:
256:
238:
216:
215:
209:
49:
771:differ at finite number of coordinates
7:
95:This article is within the scope of
920:. OK, so far. For all of the other
452:{\displaystyle E=\{0,1\}^{\omega }}
217:
38:It is of interest to the following
410:values are deducted unambiguously.
14:
1088:Low-priority mathematics articles
265:{\displaystyle \{0,1\}^{\omega }}
226:{\displaystyle 2^{\mathfrak {c}}}
115:Knowledge:WikiProject Mathematics
1083:Start-Class mathematics articles
1010:Minkowski question mark function
604:{\displaystyle x-y=(2m+1)/2^{n}}
118:Template:WikiProject Mathematics
82:
72:
51:
20:
135:This article has been rated as
962:
956:
901:
895:
583:
568:
1:
1069:08:51, 27 November 2023 (UTC)
109:and see a list of open tasks.
1052:{\displaystyle \mathbb {Q} }
1030:{\displaystyle \mathbb {D} }
715:{\displaystyle \mathbb {D} }
626:{\displaystyle \mathbb {D} }
800:{\displaystyle 2^{\omega }}
187:01:38, 4 January 2014 (UTC)
1104:
542:{\displaystyle x\approx y}
343:{\displaystyle w\approx v}
134:
67:
46:
859:pick one representative
780:Anyway, there are still
724:Baire space (set theory)
317:{\displaystyle \approx }
141:project's priority scale
700:. Third, Since the set
297:{\displaystyle \{0,1\}}
98:WikiProject Mathematics
1053:
1031:
995:
994:{\displaystyle a\in E}
969:
940:
939:{\displaystyle y\in e}
914:
913:{\displaystyle p(x)=0}
879:
878:{\displaystyle x\in e}
853:
852:{\displaystyle e\in E}
827:
826:{\displaystyle e\in E}
801:
765:
745:
716:
686:
627:
605:
543:
513:
453:
404:
384:
364:
344:
318:
298:
266:
227:
28:This article is rated
1054:
1032:
996:
970:
941:
915:
880:
854:
828:
802:
766:
746:
717:
687:
628:
606:
544:
514:
454:
405:
385:
365:
345:
319:
299:
267:
228:
1041:
1019:
979:
968:{\displaystyle p(y)}
950:
924:
889:
863:
837:
811:
784:
755:
735:
704:
641:
615:
553:
527:
470:
418:
394:
374:
354:
328:
324:defined as follows:
308:
276:
237:
208:
164:John Langford's blog
121:mathematics articles
1049:
1027:
991:
965:
936:
910:
875:
849:
823:
797:
761:
741:
712:
682:
623:
601:
539:
509:
449:
400:
380:
360:
340:
314:
294:
262:
223:
90:Mathematics portal
34:content assessment
764:{\displaystyle v}
744:{\displaystyle w}
698:quotient topology
403:{\displaystyle f}
383:{\displaystyle v}
363:{\displaystyle w}
185:
177:
155:
154:
151:
150:
147:
146:
1095:
1058:
1056:
1055:
1050:
1048:
1036:
1034:
1033:
1028:
1026:
1014:dyadic rationals
1000:
998:
997:
992:
974:
972:
971:
966:
945:
943:
942:
937:
919:
917:
916:
911:
884:
882:
881:
876:
858:
856:
855:
850:
832:
830:
829:
824:
806:
804:
803:
798:
796:
795:
770:
768:
767:
762:
750:
748:
747:
742:
721:
719:
718:
713:
711:
691:
689:
688:
683:
681:
676:
671:
670:
635:dyadic rationals
632:
630:
629:
624:
622:
610:
608:
607:
602:
600:
599:
590:
548:
546:
545:
540:
518:
516:
515:
510:
505:
500:
499:
458:
456:
455:
450:
448:
447:
409:
407:
406:
401:
389:
387:
386:
381:
369:
367:
366:
361:
349:
347:
346:
341:
323:
321:
320:
315:
303:
301:
300:
295:
271:
269:
268:
263:
261:
260:
232:
230:
229:
224:
222:
221:
220:
179:
175:
158:Machine learning
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1103:
1102:
1098:
1097:
1096:
1094:
1093:
1092:
1073:
1072:
1039:
1038:
1017:
1016:
977:
976:
948:
947:
946:, the value of
922:
921:
887:
886:
861:
860:
835:
834:
809:
808:
787:
782:
781:
753:
752:
733:
732:
702:
701:
662:
639:
638:
613:
612:
591:
551:
550:
525:
524:
491:
468:
467:
439:
416:
415:
392:
391:
372:
371:
352:
351:
326:
325:
306:
305:
274:
273:
252:
235:
234:
211:
206:
205:
202:axiom of choice
194:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1101:
1099:
1091:
1090:
1085:
1075:
1074:
1047:
1037:and rationals
1025:
990:
987:
984:
964:
961:
958:
955:
935:
932:
929:
909:
906:
903:
900:
897:
894:
874:
871:
868:
848:
845:
842:
822:
819:
816:
794:
790:
760:
740:
710:
694:quotient space
680:
675:
669:
665:
661:
658:
655:
652:
649:
646:
633:is the set of
621:
598:
594:
589:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
538:
535:
532:
508:
504:
498:
494:
490:
487:
484:
481:
478:
475:
446:
442:
438:
435:
432:
429:
426:
423:
412:
411:
399:
379:
359:
339:
336:
333:
313:
293:
290:
287:
284:
281:
259:
255:
251:
248:
245:
242:
219:
214:
193:
190:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1100:
1089:
1086:
1084:
1081:
1080:
1078:
1071:
1070:
1066:
1062:
1015:
1011:
1007:
1002:
988:
985:
982:
959:
953:
933:
930:
927:
907:
904:
898:
892:
872:
869:
866:
846:
843:
840:
820:
817:
814:
792:
788:
778:
776:
772:
758:
738:
729:
725:
699:
695:
673:
667:
659:
656:
653:
647:
644:
636:
596:
592:
587:
580:
577:
574:
571:
565:
562:
559:
556:
536:
533:
530:
522:
506:
502:
496:
488:
485:
482:
476:
473:
466:
462:
444:
436:
433:
430:
424:
421:
397:
377:
357:
337:
334:
331:
311:
288:
285:
282:
257:
249:
246:
243:
212:
203:
199:
198:
197:
191:
189:
188:
183:
178:
171:
170:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1061:67.198.37.16
1003:
779:
775:cylinder set
731:
728:Cantor space
465:quotient set
461:Cantor space
413:
195:
167:
161:
137:Low-priority
136:
96:
62:Lowβpriority
40:WikiProjects
169:Perceptrons
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1077:Categories
1006:Vitali set
885:, and set
519:with this
773:(aka the
200:Assuming
521:cofinite
176:VVERTYVS
692:as the
139:on the
36:scale.
1065:talk
751:and
370:and
350:if
272:to
182:hm?
131:Low
1079::
1067:)
986:β
931:β
870:β
844:β
818:β
793:Ο
668:Ο
560:β
534:β
507:β
497:Ο
445:Ο
335:β
312:β
258:Ο
1063:(
1046:Q
1024:D
989:E
983:a
963:)
960:y
957:(
954:p
934:e
928:y
908:0
905:=
902:)
899:x
896:(
893:p
873:e
867:x
847:E
841:e
821:E
815:e
789:2
759:v
739:w
709:D
679:D
674:/
664:}
660:1
657:,
654:0
651:{
648:=
645:E
620:D
597:n
593:2
588:/
584:)
581:1
578:+
575:m
572:2
569:(
566:=
563:y
557:x
537:y
531:x
503:/
493:}
489:1
486:,
483:0
480:{
477:=
474:E
441:}
437:1
434:,
431:0
428:{
425:=
422:E
398:f
378:v
358:w
338:v
332:w
292:}
289:1
286:,
283:0
280:{
254:}
250:1
247:,
244:0
241:{
218:c
213:2
184:)
180:(
174:Q
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
