FUZZY RELATIONS and COMPOSITION OF FUZZY RELATIONS

Transcription

1 Fu eltos FUZZ ELIONS d COMPOSIION OF FUZZ ELIONS Fu relto geerles clsscl relto to oe tht llows prtl membershp d descrbes reltoshp tht holds betwee two or more objects. Emple: fu relto Fred descrbe the degree of fredshp betwee two persos cotrst to ether beg fred or ot beg fred clsscl relto! Fu Logc:Itellgece Cotrol d Iformto J. e d. Lgr PretceHll Fu dvce Crsp Crtes product Lets cosder propertes of crsp reltos frst d the eted the mechsm to fu sets. Defto of crsp Product set: Let d B be two o-empt sets the product set or Crtes product B s defed s follows B { b b B } Bewre of the mth! set of ordered prs b Crsp Crtes product Crsp Br elto Crtes product of sets... { } Defto of Br elto If d B re two sets d there s specfc propert betwee elemets of d of B ths propert c be descrbed usg the ordered pr. set of such prs d B s clled relto. { B }

2 Crsp Br elto Emples of Br reltos: Crsp -r elto Defto of -r relto For sets... the relto mog elemets... c be descrbed b -tuple.... collecto of such -tuples... s relto mog Let Fu Crtes Product be membershp fuctos of for. he the Crtes product the probblt for -tuple to be volved fu set s m[ ] Fu Crtes Product: Emple Let be fu set defed o uverse of three dscrete tempertures X { } d B be fu set defed o uverse of two dscrete pressures { } Fu set represets the mbet temperture d fu set B the er optmum pressure for cert het echger d the Crtes product mght represet the codtos temperture-pressure prs of the echger tht re ssocted wth effcet opertos. For emple let. B..9 Fu Logc wth Egeerg pplctos: moth J. oss McGrw-Hll. B....9 Fu elto Fu elto fu relto s mppg from the Crtes spce X to the tervl [] where the stregth of the mppg s epressed b the membershp fucto of the relto : B [ ] { B}

3 Fu elto Crsp relto vs. Fu relto Correspodg fu relto mtr Composto of Fu eltos wo fu reltos d S re defed o sets B d C. ht s B S B C. he composto S S of two reltos d S s epressed b the relto from to C: For B B C S m [m S ] [ S ] M S M M S mtr otto m-m composto Composto of Fu eltos Composto of Fu eltos Emple: { X } { } S m[m.9m.] d Z { } Cosder the followg fu reltos:.8 d S.9. Usg m-m composto }.8 Fu Logc wth Egeerg pplctos: moth J. oss McGrw-Hll Composto of Fu eltos wo fu reltos d S re defed o sets B d C. ht s B S B C. he composto S S of two reltos d S s epressed b the relto from to C: For B B C S m [ S ] [ S ] M S M M S mtr otto m-product composto Composto of Fu eltos M-product emple: X } { } { d Z { } Cosder the followg fu reltos:.8 d S.9. Usg m-product composto S } m[.8 ] 8. 8 Fu Logc wth Egeerg pplctos: moth J. oss McGrw-Hll

4 pplcto: Computer Egeerg Problem: I computer egeerg dfferet logc fmles re ofte compred o the bss of ther power-del product. Cosder the fu set F of logc fmles the fu set D of del tmes s d the fu set P of power dssptos mw. If F {NMOSCMOSLECLJJ} D {.} P {..} Suppose D F d F P.. N C E J N.. C.. d E. J. Fu Logc wth Egeerg pplctos: moth J. oss McGrw-Hll pplcto: Computer Egeerg Cot We c use m-m composto to obt relto betwee del tmes d power dsspto:.e. we c compute or Fu Logc wth Egeerg pplctos: moth J. oss McGrw-Hll pplcto: Fu elto Pette Fu elto Pette defes the degree b whch perso wth specfc heght d weght s cosdered pette. Suppose the rge of the heght d the weght of terest to us re { } d { } lb. We c epress the fu relto mtr form s show below: ".9.. ". P ". 4".8. 5". 6" pplcto: Fu elto Pette Oce we defe the pette fu relto we c swer two kds of questos: Wht s the degree tht femle wth specfc heght d specfc weght s cosdered to be pette? Wht s the possblt tht pette perso hs specfc pr of heght d weght mesures? ".9.. ". P ". 4".8. 5". 6" Fu Logc:Itellgece Cotrol d Iformto J. e d. Lgr PretceHll Fu Logc:Itellgece Cotrol d Iformto J. e d. Lgr PretceHll pplcto: Fu elto Pette Gve two-dmesol fu relto d the possble vlues of oe vrble fer the possble vlues of the other vrble usg smlr fu composto s descrbed erler. Defto: Let X d be the uverses of dscourse for vrbles d respectvel d d j be elemets of X d. Let be fu relto tht mps X to [] d the possblt dstrbuto of X s kow to be P. he compostol rule of ferece fers the possblt dstrbuto of s follows: m-m composto: m-product composto: mm j X j Fu Logc:Itellgece Cotrol d Iformto J. e d. Lgr PretceHll j m X j P pplcto: Fu elto Pette Problem: We m wsh to kow the possble weght of pette femle who s bout 5 4. ssume bout 5 4 s defed s bout-5 4 {/5 /5 /5.8/5 /5 4.8/5 5 /5 6 } Usg m-m compostol we c fd the weght possblt dstrbuto of pette perso bout 5 4 tll: ".9.. ". ". 4".8. 5". 6" weght Smlrl we c compute the possblt degree for other weghts. he fl result s weght {.8 / 9.8 / 95.8 /.8/ 5 / /5./ /5} Fu Logc:Itellgece Cotrol d Iformto J. e d. Lgr PretceHll 4

5 Fu Grphs fu relto m ot hve megful lgustc lbel. Most fu reltos used rel-world pplctos do ot represet cocept rther the represet fuctol mppg from set of put vrbles to oe or more output vrbles. Fu rules c be used to descrbe fu relto from the observed stte vrbles to cotrol decso usg fu grphs fu grph descrbes fuctol mppg betwee set of put lgustc vrbles d output lgustc vrble. Fu Logc:Itellgece Cotrol d Iformto J. e d. Lgr PretceHll 5