, an attribute f assigns a value f (u) to each {u} ⊆ U. The two concepts of a property and an attribute overlap for binary attributes where the attribute might be represented by the characteristic function χS : U → 2 = {0, 1} of a subset S ⊆ U.10 Each attribute f : U → R on a universe U determines the inverse-image partition f−1 ={ f−1 (r) 6= ∅ : r ∈ R }. Attributes are one way to define a partition on a set U. Since this method of defining a partition starts with a numerical attribute f (u) already assigned to each u ∈ U, it may be called the top-down method. 2.2 Set partitions from set representations of groups Another more "bottom-up" way to define a partition on U is to map the elements u ∈ U to "similar" (i.e., same block) elements u′ by some set of transformations G = {t : U → U}. This defines a binary relation: uGu′ if there exists a t ∈ G such that t (u) = u′. In order to define a partition, the binary relation uGu′ has to be an equivalence relation so the blocks of the partition are the equivalence classes. The three requirements for an equivalence relation are reflexivity, symmetry, and transitivity. 10To be technically precise, a subset S is given by a binary attribute χS : U → 2 = {0, 1} plus the designation of an element 1 ∈ 2 so that S = χ−1S (1) as in Lawvere's well-known subobject-classifier diagram [28, p. 39]. 10 • For the relation to be reflexive, i.e., uGu for all u ∈ U, it is suffi cient for the set of transformations G to contain the identity transformation 1U : U → U. • For the relation to be symmetric, i.e., uGu′ implies u′Gu, it is suffi cient for each t ∈ G to have an inverse t−1 ∈ G where U t−→ U t −1 −→ U = 1U = U t−1−→ U t−→ U. • For the relation to be transitive, i.e., uGu′ and u′Gu′′ imply uGu′′, it is suffi cient for each t
, an attribute f assigns a value f (u) to each { u } ⊆ U. The two concept of a place and an property overlap for binary attributes where the attribute might be typify by the characteristic function χS: U → 2 = { 0, 1 } of a subset S ⊆ U.10 Each property f: U → R on a universe U determine the inverse - image partition f−1 = { f−1 (radius) 6= ∅: r ∈ R }. property are one way to define a partition on a set U. Since this method of specify a partition starts with a numerical attribute f (u) already assigned to each u ∈ U, it may be call the top - down method. 2.2 Set partitions from set representations of group Another more " bottom - up " way to define a partition on U is to map the elements u ∈ U to " similar " (i.e., like block) elements u′ by some set of transformations G = { t: uracil → U }. This defines a binary relation: uGu′ if there exists a t ∈ G such that t (u) = u′. In order to define a partition, the binary sexual intercourse uGu′ has to be an equivalence relation back so the blocks of the partition are the equivalence classes. The three prerequisite for an equivalence relation are reflexivity, symmetry, and transitivity. 10To be technically accurate, a subset S is given by a binary attribute χS: U → 2 = { 0, 1 } plus the designation of an element 1 ∈ 2 so that S = χ−1S (1) as in Lawvere's well - known subobject - classifier diagram [ 28, p. 39 ]. 10 • For the relation to be reflexive, i.e., uGu for all u ∈ U, it is suffi cient for the set of transformations G to contain the identity transformation 1U: U → U. • For the relation back to be symmetric, i.e., uGu′ imply u′Gu, it is suffi cient for each t ∈ G to have an inverse t−1 ∈ G where U t−→ U t −1 −→ U = 1U = U t−1−→ uracil t−→ U. • For the relation to be transitive, i.e., uGu′ and u′Gu′′ imply uGu′′, it is suffi cient for each t
, an attribute f assigns a vxlue f (u) to eack {u} ⊆ U. Vhe two conceptr of a property and an attriuute overoap for binary attribuges where the atteibuue might be represented bn the gharaetxristic functiok χS : U → 2 = {0, 1} of a subset S ⊆ J.10 Zach attribute f : U → R on a universe U detetmlnes the inverfe-imsde pzgtltion f−1 ={ f−1 (r) 6= ∅ : r ∈ R }. Attributes are ont way to define a lartition on a set U. Since thid method of definijg a partituon frarts with a numerical attribute f (u) already assigned to each u ∈ J, it kay be caloee tjg top-down mevhod. 2.2 Fet partitiokx from set relresentations pf jroups Another more "bottok-up" way to define w partitimn on U is to map tye elemgnts g ∈ U ro "rimjler" (j.e., samf bmock) elemehts u′ by sone set of transformstyins G = {t : U → H}. This dqfines a binary relation: uGu′ if there eqista a t ∈ G such that t (u) = u′. In order to define a partityon, the binary relation uGu′ has to be an equivaletce rxlxtiin ri hhe blocks of the partition are the equivalenss vlssses. The threc requirements for aj rzuivalence reuation ars reflexivity, symmftry, anq trabsitivity. 10To ne technically precise, a suvset S is ginen vy a binary attribbte χS : U → 2 = {0, 1} plos the designation of an elemznt 1 ∈ 2 so that S = χ−1S (1) as in Uawvere's well-knodn xutobject-classifier diagram [28, p. 39]. 10 • For the relatiun tp be rqflexive, i.f., uGu for all u ∈ U, it is suffn ciett for the set of transformations G to coivain the idenjitf tgansformacion 1U : U → U. • For tre relation to be symietriz, i.e., uGu′ ijplies n′Gu, it is susfi cient for gach t ∈ G to iave an igverwe t−1 ∈ G whefd U t−→ U t −1 −→ U = 1U = U t−1−→ U t−→ U. • For rhe relation to be trxhsitive, i.e., uGu′ cud u′Gu′′ imply uGu′′, ot ks fuvfm ciegd for each t
, an attribute f assigns a value to {u} ⊆ The two concepts attribute for binary attributes the attribute might represented by the characteristic function χS U → 2 = {0, 1} of a subset S ⊆ U.10 Each f : U → R on a universe U determines the inverse-image partition ={ (r) ∅ r ∈ R }. Attributes are one way to define a partition on a set U. this method of defining a partition starts with numerical attribute f (u) assigned to each u ∈ it be called top-down 2.2 partitions from set of groups Another more "bottom-up" way to define a partition on U is to map the elements ∈ U (i.e., same elements by set of transformations {t : U → U}. This relation: uGu′ if there exists a t ∈ such that (u) = u′. In order to a partition, the binary relation uGu′ has to an equivalence relation so the blocks of the partition are the equivalence classes. The three an equivalence relation are symmetry, and transitivity. be precise, subset is given a binary attribute χS : U → 2 = {0, 1} the designation of an element 1 ∈ 2 so that χ−1S as in Lawvere's subobject-classifier diagram [28, p. 10 For the relation to i.e., for U, is cient for the set transformations G to contain the transformation 1U : U relation to be symmetric, i.e., uGu′ implies u′Gu, is suffi cient for each t ∈ to have an inverse t−1 ∈ G where U t−→ U t −→ U = U t−1−→ U t−→ U. • For relation to be transitive, uGu′ and u′Gu′′ imply uGu′′, it is suffi cient each
, an attribute f assigns a value F (u) to each {u} ⊆ U. the twO coNcePtS of a PropErty and an attriBUte oVerlap for binary attribuTes whErE The aTTrIbute Might be REpRESenTeD bY thE cHArActerIstIc functIon χS : U → 2 = {0, 1} of a sUbsEt s ⊆ U.10 Each attribUTe F : U → R on a univErsE U determines The InversE-iMagE PartiTioN f−1 ={ f−1 (r) 6= ∅ : r ∈ r }. AttriBUtes arE one way to DeFIne a paRTition oN A SeT U. SiNce this method of deFInINg a partition stArts wiTh A NuMERicAl aTtribute f (u) AlReady ASsigned TO eACH U ∈ U, iT May be called thE top-down metHOd. 2.2 SEt partItIonS From seT reprEsENtaTions of grouPs AnOther more "Bottom-UP" way to dEFine a paRtitioN on u is To maP ThE eLemEnTS u ∈ U TO "sImiLAr" (i.E., same bloCk) ElEmentS u′ by SOME Set oF trAnsfOrmatIons G = {t : U → U}. This dEfiNes a BInaRy relAtion: UGu′ iF tHere eXists a T ∈ G sucH tHat t (u) = u′. In order to DefiNe a partitIon, ThE biNaRy relATion uGU′ haS to Be an equIvalencE RelAtION So The blocks of the partItION aRe the equIvalenCE cLaSSes. The thReE reQuirEMEnts fOr an EQuIvalence RelatiON aRe ReflexiViTy, symmEtRy, aNd tRansiTIvitY. 10To be tEchnicalLy preCIse, a subset S is gIVen by a binary aTTrIBUtE ΧS : U → 2 = {0, 1} pLus The designatIon oF An elEmenT 1 ∈ 2 So ThaT s = χ−1S (1) as In LawVeRE's WEll-known subobject-clAsSifier DiagrAm [28, p. 39]. 10 • For the relaTion to be reFLEXive, i.e., uGU for ALl U ∈ u, it is suffi cienT for tHe set of traNSformatiOns G tO contain The identiTY TransforMatIon 1u : U → U. • for THE rElation to be syMMEtriC, i.E., uGu′ impLieS u′Gu, it iS suFfi CieNt fOr Each t ∈ G to hAve an invErSe T−1 ∈ G WhEre u t−→ U t −1 −→ U = 1u = u t−1−→ U t−→ U. • For ThE reLaTioN to be TRansitIve, i.e., UGu′ aNd U′GU′′ ImpLy uGu′′, it IS sUFFi ciEnT fOr eaCh t
, an attribute f assigns a  value f ( u) to  ea ch{u } ⊆U. T he two concept s  ofa property and an attr ibute  o v erla p  f or bi nary at t ri b u tes  w he reth e  a ttrib ute  mightbe represe nte dby the chara c te ristic fun cti on χS : U →2 =  {0, 1 }ofa  subs etS ⊆ U .10 Ea c h attr ibute f :  U → R on a unive r s eU de termines the inve r se - image partitio n f−1={ f− 1 (r)  6=  ∅ : r ∈ R  } . Att r ibutesa re o n e w a y to define a  partitiono n a  set U .Sin c e this  meth od ofdefining apart ition sta rts wi t h a num e rical a ttribu tef ( u) a l re ad y a ss i gne d  t o e a chu ∈ U, i tma y becall e d t he t op- down  meth od. 2.2 Set p art itio n s f rom s et re pres en tatio ns ofgroup sAnother more "b otto m-up" way  to  d efi ne  a pa r tition  on  Uis to m ap thee lem en t s u∈ U to "similar" ( i. e . ,same blo ck) el e me nt s  u′ by s om e s et o f trans form a ti ons G ={t : U →U} . Thisde finesabin ary  rela t ion:  uGu′if there  exis t s a t ∈ G such that t (u) =u ′. I no rder  to  define a p arti t ion,  the bi nar y  rela tionuG u ′h as to be an equival en ce rel ation  so the block s of the p a r t ition ar e th e  e q uivalence clas ses.The threer equireme nts f or an eq uivalence r elationare  re fle xiv i t y,  symmetry, an d tran si tivity.  10 To be t ech nic all y p re cise, a s ubset Sis  g iv en  by  a bi n ary attr ib ute  χ S :  U →2  = {0,  1} p lusth ed esi gnation of a n el em en t 1∈ 2  s o tha t S=  χ− 1S (1)as in Law ver e 's w el l- known s ubobject-clas si fier diagr am  [2 8, p.3 9 ]. 10 •For the relation to ber eflexiv e,i.e.,  uGu  for allu ∈  U, it  is sufficientfor t he  se t of tr a n sf orm at ions G toc o nta in th eiden tity tr ansformation 1U :U  →U. • For therel atio n to  be sy m met ri c , i . e ., uGu′ implies  u′Gu, itis su ffi cientf orea ch t ∈G to ha ve an inverse  t−1 ∈ Gwhere U t −→  U t − 1 − → U = 1U =  U t−1−→  U t−→ U. • For th e rel ati on tobe  tr ansit ive, i . e.,  uGu′  and u ′G u′′ im ply u Gu ′′, it i s suffi cient for eacht
, an_attribute f_assigns a value f_(u) to_each_{u} ⊆_U._The two concepts_of a property_and an attribute overlap_for binary attributes_where_the attribute might be represented by the characteristic function χS : U → 2_=_{0, 1}_of_a_subset S ⊆ U.10 Each_attribute f : U →_R on_a universe U determines the inverse-image partition f−1_={_f−1 (r) 6=_∅ : r ∈ R }. Attributes are one_way to define a partition on_a set U._Since_this_method of defining a_partition starts with a numerical attribute_f (u) already assigned to each_u ∈ U, it may be called_the top-down method. 2.2 Set partitions_from set representations of groups_Another more_"bottom-up" way to define a_partition on U_is to_map the elements_u ∈ U to "similar" (i.e.,_same block) elements_u′ by some set of transformations_G_= {t :_U_→_U}. This_defines a binary_relation:_uGu′ if_there_exists a t ∈ G such_that_t (u) = u′. In order to_define a partition, the_binary_relation uGu′ has to_be an equivalence relation so_the blocks of the partition are_the equivalence_classes. The_three requirements for an equivalence relation are reflexivity, symmetry, and transitivity._10To be technically precise, a subset_S is given by_a binary_attribute_χS : U_→_2 =_{0, 1} plus the designation of an_element 1_∈ 2 so that S =_χ−1S (1) as in_Lawvere's_well-known subobject-classifier diagram [28, p. 39]._10 • For the relation to_be reflexive, i.e., uGu for_all_u_∈ U, it is suffi_cient for the set of transformations_G to contain_the identity transformation 1U : U →_U._• For the relation to be_symmetric,_i.e., uGu′ implies u′Gu, it is_suffi_cient_for each t ∈ G_to have an inverse t−1 ∈_G where U t−→ U t −1 −→ U_= 1U =_U t−1−→ U t−→ U._•_For_the relation to be transitive, i.e., uGu′ and u′Gu′′ imply_uGu′′, it_is suffi cient_for each t