17:
48:(of which the set in question is a subset), such that each partition can be mapped back onto the entire set using only finitely many distinct functions (or compositions thereof) to accomplish the mapping. A set that admits such a paradoxical decomposition where the actions belong to a group
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24:
is that a ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original.
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40:. A paradoxical decomposition of a set is two families of disjoint subsets, along with appropriate
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115:. Admitting infinite classes as sets is sufficient to allow paradoxical sets.
29:
665:{\displaystyle F=\{e\}\cup X(a)\cup X(a^{-1})\cup X(b)\cup X(b^{-1})}
705:
is the collection of all (reduced) words that start with the letter
15:
811:{\displaystyle X(a)\cup aX(a^{-1})=F=X(b)\cup bX(b^{-1}).}
292:{\displaystyle A_{1},...,A_{n},B_{1},...,B_{m}\subseteq A}
835:, which divides the sphere into paradoxical sets for the
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388:{\displaystyle g_{1},...,g_{n},h_{1},...,h_{m}\in G}
831:The most famous example of paradoxical sets is the
203:-paradoxical if there exists some disjoint subsets
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530:{\displaystyle A=\bigcup _{i=1}^{m}h_{i}(B_{i})}
460:{\displaystyle A=\bigcup _{i=1}^{n}g_{i}(A_{i})}
877:(Second ed.). Cambridge University Press.
111:Paradoxical sets exist as a consequence of the
709:. This is a paradoxical decomposition because
8:
579:
573:
88:-paradoxical or paradoxical with respect to
871:Wagon, Stan; Tomkowicz, Grzegorz (2016).
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1:
839:. This result depends on the
925:
853:Pathological (mathematics)
824:
874:The Banach–Tarski Paradox
676:is the identity word and
38:paradoxical decomposition
837:special orthogonal group
299:and some group elements
812:
699:
666:
560:has the decomposition
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461:
430:
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197:
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833:Banach–Tarski paradox
827:Banach–Tarski paradox
821:Banach–Tarski paradox
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22:Banach–Tarski paradox
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909:Geometric dissection
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698:{\displaystyle X(i)}
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36:is a set that has a
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556:on two generators
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26:
884:978-1-107-04259-9
196:{\displaystyle G}
176:{\displaystyle A}
156:{\displaystyle A}
136:{\displaystyle G}
113:Axiom of Infinity
101:{\displaystyle G}
81:{\displaystyle G}
61:{\displaystyle G}
44:that act on some
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123:Suppose a group
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841:axiom of choice
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34:paradoxical set
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825:Main article:
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143:acts on a set
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55:
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43:
42:group actions
39:
35:
31:
23:
18:
873:
866:
830:
706:
673:
557:
553:
548:
397:
122:
110:
37:
33:
27:
395:such that:
904:Set theory
898:Categories
859:References
551:Free group
545:Free group
119:Definition
68:is called
30:set theory
795:−
778:∪
746:−
729:∪
652:−
638:∪
623:∪
612:−
598:∪
583:∪
482:⋃
412:⋃
380:∈
284:⊆
847:See also
540:Examples
46:universe
163:. Then
881:
672:where
879:ISBN
549:The
467:and
32:, a
20:The
558:a,b
183:is
28:In
900::
843:.
108:.
887:.
806:.
803:)
798:1
791:b
787:(
784:X
781:b
775:)
772:b
769:(
766:X
763:=
760:F
757:=
754:)
749:1
742:a
738:(
735:X
732:a
726:)
723:a
720:(
717:X
707:i
693:)
690:i
687:(
684:X
674:e
660:)
655:1
648:b
644:(
641:X
635:)
632:b
629:(
626:X
620:)
615:1
608:a
604:(
601:X
595:)
592:a
589:(
586:X
580:}
577:e
574:{
571:=
568:F
554:F
525:)
520:i
516:B
512:(
507:i
503:h
497:m
492:1
489:=
486:i
478:=
475:A
455:)
450:i
446:A
442:(
437:i
433:g
427:n
422:1
419:=
416:i
408:=
405:A
383:G
375:m
371:h
367:,
364:.
361:.
358:.
355:,
350:1
346:h
342:,
337:n
333:g
329:,
326:.
323:.
320:.
317:,
312:1
308:g
287:A
279:m
275:B
271:,
268:.
265:.
262:.
259:,
254:1
250:B
246:,
241:n
237:A
233:,
230:.
227:.
224:.
221:,
216:1
212:A
191:G
171:A
151:A
131:G
96:G
76:G
56:G
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