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simravleebi, funqciebi, mimarTebebi leqcia # 1 PowerPoint PPT Presentation


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logika da diskretuli maTematika Tbilisis saxelmwifo universiteti zusti da sabunebismetyvelo mecnierebaTa fakulteti informatikis mimarTuleba bakalavriati prof. revaz grigolia. simravleebi, funqciebi, mimarTebebi leqcia # 1. 1.1 simravluri operaciebi gaerTianeba  da TanaukveTi gaerTianeba

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simravleebi, funqciebi, mimarTebebi leqcia # 1

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logika da diskretuli maTematikaTbilisis saxelmwifo universitetizusti da sabunebismetyvelo mecnierebaTa fakultetiinformatikis mimarTulebabakalavriatiprof. revaz grigolia


simravleebi, funqciebi, mimarTebebileqcia # 1

  • 1.1 simravluri operaciebi

  • gaerTianeba da TanaukveTi gaerTianeba

    • TanakveTa 

    • sxvaoba “-”

    • damateba“”

    • simetriuli sxvaoba 


simravleebi, funqciebi, mimarTebebileqcia # 1universaluri simravle

roca Cven vsaubrobT simravleebze, universaluri simravle saWiroebs dazustebas. miuxedavad imisa, rom simravle ganisazRvreba misi elementebiT, romlebsac is Seicavs, es elementebi ar SeiZleba iyos nebismieri. Tu nebismier elementebis Semcveloba daSvebulia, maSin SesaZlebelia miviRoT paradoqsebi TvTmiTiTebis xarjze.


simravleebi, funqciebi, mimarTebebileqcia # 1

rasselis paradoqsi

Tu Cven davuSvebT nebismier elements, maSin Cven unda davuSvaT agreTve, rom simravlec iyos elementi, agreTve simravleebis simravlec, da a. S. amitom, savsebiT dasaSvebia ganvixiloT Semdegi simravle:

S = simravle Semdgari yvela im simravleebisagan,

romlebic ar Seicaven Tavis Tavs.

winadadeba. es simravle ar arsebobs.


simravleebi, funqciebi, mimarTebebileqcia # 1

rasselis paradoqsi

damtkiceba. Tu es simravle arsebobs, maSin is an unda Seicavdes Tavis Tavs an ar unda Seicavdes. ganvixiloT orive SemTveva:

1.S-i Seicavs Tavis Tavs, rogorc elements. amitom, vinaidan S-is elementebi ar Seicaven Tavis Tavs, elementis saxiT, maSin is ar Seicavs S-s. es ki ewinaamRdegeba daSvebas, da amitom es pirveli SemTveva SeuZlebelia.


simravleebi, funqciebi, mimarTebebileqcia # 1

rasselis paradoqsi

2.S-i ar Seicavs Tavis Tavs, rogorc elements. amitom, vinaidan S-i Seicavs yvela im simravleebs, romlebic ar Seicaven Tavis Tavs, elementis saxiT, maSin is unda Seicavdes S-s. es ki ewinaamRdegeba daSvebas, da amitom meore SemTxvevac SeuZlebelia.

vinaidan orive SemTxveva SeuZlebelia, S-i ar arsebobs. 


simravleebi, funqciebi, mimarTebebileqcia # 1

  • {x  N | y (x = 2y ) }

  • {0,1,8,27,64,125, …}

    kiT1: U = N. { x | y (y  x ) } = ?

    kiT2: U = Z. { x | y (y  x ) } = ?

    kiT3: U = Z. { x | y (y  R y 2 = x )} = ?

    kiT4: U = Z. { x | y (y  R y 3 = x )} = ?

    kiT5: U = R. { |x | | x  Z } = ?

    kiT6: U = R. { |x | } = ?


simravleebi, funqciebi, mimarTebebileqcia # 1

pas1: U = N. { x | y (y  x ) } = { 0 }

pas2: U = Z. { x | y (y  x ) } = { }

pas3: U = Z. { x |y (y  R y 2 = x )}

= { 0, 1, 2, 3, 4, … } = N

pas4: U = Z. { x |y (y  R y 3 = x )} = Z

pas5: U = R. { |x | | x  Z } = N

pas6: U = R. { |x | } = arauaryofiTi

namdvili ricxvebi.


simravleebi, funqciebi, mimarTebebileqcia # 1

Semdegi Teoriul-simravluri operaciebi

  • gaerTianeba ()

  • TanakveTa ()

  • sxvaoba (-)

  • damateba (“—”)

  • simetriuli sxvaoba ()

    gvaZleven simravleebs: AB, AB, A-B, AB, andA.


simravleebi, funqciebi, mimarTebebileqcia # 1gaerTianeba

  • elementebi ekuTvnis ori simravlidan erTs mainc

AB = { x | x  A  x  B }

U

AB

A

B


TanakveTa

simravleebi, funqciebi, mimarTebebi

  • elementebi ekuTvnis zustad orive simravles

AB = { x | x  A  x  B }

U

A

B

AB


simravleebi, funqciebi, mimarTebebiTanaukveTi simravleebi

  • gans: TuAdaB-sar gaaCniaT saerTo elementebi, maSin mas uwodeben TanaukveT simravleebs,e. i. A B =  .

U

A

B


simravleebi, funqciebi, mimarTebebiTanaukveTi gaerTianeba

U

A

B


simravleebi, funqciebi, mimarTebebisxvaoba

  • elementebi ekuTvnis pirvel simravles da ara meores

A-B = { x | x  A  x  B }

U

A-B

B

A


simravleebi, funqciebi, mimarTebebisimetriuli sxvaoba

  • elementebi ekuTvnis oridan mxolod erT simravles

AB = { x | x  A  x  B }

AB

U

A

B


simravleebi, funqciebi, mimarTebebidamateba

  • elementebi ar ekuTvnis simravles (unaruli operatoria)

A = { x | x  A }

U

A

A


simravleebi, funqciebi, mimarTebebisimravluri igiveobebi

  • logikuri igiveobebi hqmnian simravlur igiveobebs sxvadasxva simravluri operaciebis gansazRvris gamoyenebiT. magaliTad:

  • lema. (gaerTianebis asociaciuroba).

    (AB )C = A(B C )


simravleebi, funqciebi, mimarTebebisimravluri igiveobebi

  • logikuri igiveobebi hqmnian simravlur igiveobebs sxvadasxva simravluri operaciebis gansazRvris gamoyenebiT. magaliTad:

  • lema. (gaerTianebis asociaciuroba).

    (AB )C = A(B C )

    damtkiceba. (AB )C = {x | x  A B  x  C }(gans. Tan.)


simravleebi, funqciebi, mimarTebebisimravluri igiveobebi

  • logikuri igiveobebi hqmnian simravlur igiveobebs sxvadasxva simravluri operaciebis gansazRvris gamoyenebiT. magaliTad:

  • lema. (gaerTianebis asociaciuroba).

    (AB )C = A(B C )

    damtkiceba. (AB )C = {x | x  A B  x  C } (gans. Tan.)

    = {x |(x  A  x  B )  x  C }(gans. Tan.)


simravleebi, funqciebi, mimarTebebisimravluri igiveobebi

  • logikuri igiveobebi hqmnian simravlur igiveobebs sxvadasxva simravluri operaciebis gansazRvris gamoyenebiT. magaliTad:

  • lema. (gaerTianebis asociaciuroba).

    (AB )C = A(B C)

    damtkiceba. (AB )C = {x | x  A B  x  C } (gans. Tan.)

    = {x |(x  A  x  B )  x  C }(gans. Tan.)

    = {x | x  A  ( x  B  x  C ) } (log. asoc.)


simravleebi, funqciebi, mimarTebebisimravluri igiveobebi

  • logikuri igiveobebi hqmnian simravlur igiveobebs sxvadasxva simravluri operaciebis gansazRvris gamoyenebiT. magaliTad:

  • lema. (gaerTianebis asociaciuroba).

    (AB )C = A(B C)

    damtkiceba. (AB )C = {x | x  A B  x  C } (gans. Tan.)

    = {x |(x  A  x  B )  x  C }(gans. Tan.)

    = {x | x  A  ( x  B  x  C ) } (log. asoc.)

    = {x | x  A  x  B  C ) } (gans. Tan.)


simravleebi, funqciebi, mimarTebebisimravluri igiveobebi

  • logikuri igiveobebi hqmnian simravlur igiveobebs sxvadasxva simravluri operaciebis gansazRvris gamoyenebiT. magaliTad:

  • lema. (gaerTianebis asociaciuroba).

    (AB )C = A(B C)

    damtkiceba. (AB )C = {x | x  A B  x  C } (gans. Tan.)

    = {x |(x  A  x  B )  x  C }(gans. Tan.)

    = {x | x  A  ( x  B  x  C ) } (log. asoc.)

    = {x | x  A  x  B  C ) } (gans. Tan.)

    = A(B C ) (gans. Tan.) 

    analogiurad gamoyvaneba sxva igiveobebi.


simravleebi, funqciebi, mimarTebebisimravluri igiveobebi venis diagramebis saSualebiT

  • xSirad ufro advilia simravluri igiveobebis gageba venis diagramebis daxatviT.

  • magaliTad ganvixiloT de morganis pirveli kanoni


simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:


simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

AB:


simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

AB:


simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:


simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

A:

B:


simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

A:

B:


simravleebi, funqciebi, mimarTebebide morganis kanoni


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