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


Simravleebi funqciebi mimartebebi leqcia 1
simravleebi, funqciebi, mimarTebebileqcia # 1

  • 1.1 simravluri operaciebi

  • gaerTianeba da TanaukveTi gaerTianeba

    • TanakveTa 

    • sxvaoba “-”

    • damateba“”

    • simetriuli sxvaoba 


Simravleebi funqciebi mimartebebi leqcia 1 universaluri simravle
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 mimartebebi leqcia 11
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 mimartebebi leqcia 12
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 mimartebebi leqcia 13
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 mimartebebi leqcia 14
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 mimartebebi leqcia 15
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 mimartebebi leqcia 16
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 mimartebebi leqcia 1 gaertianeba
simravleebi, funqciebi, mimarTebebileqcia # 1gaerTianeba

  • elementebi ekuTvnis ori simravlidan erTs mainc

AB = { x | x  A  x  B }

U

AB

A

B


Simravleebi funqciebi mimartebebi

TanakveTa

simravleebi, funqciebi, mimarTebebi

  • elementebi ekuTvnis zustad orive simravles

AB = { x | x  A  x  B }

U

A

B

AB


Simravleebi funqciebi mimartebebi tanaukveti simravleebi
simravleebi, funqciebi, mimarTebebiTanaukveTi simravleebi

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

U

A

B


Simravleebi funqciebi mimartebebi tanaukveti gaertianeba
simravleebi, funqciebi, mimarTebebiTanaukveTi gaerTianeba

U

A

B


Simravleebi funqciebi mimartebebi sxvaoba
simravleebi, funqciebi, mimarTebebisxvaoba

  • elementebi ekuTvnis pirvel simravles da ara meores

A-B = { x | x  A  x  B }

U

A-B

B

A


Simravleebi funqciebi mimartebebi simetriuli sxvaoba
simravleebi, funqciebi, mimarTebebisimetriuli sxvaoba

  • elementebi ekuTvnis oridan mxolod erT simravles

AB = { x | x  A  x  B }

AB

U

A

B


Simravleebi funqciebi mimartebebi damateba
simravleebi, funqciebi, mimarTebebidamateba

  • elementebi ar ekuTvnis simravles (unaruli operatoria)

A = { x | x  A }

U

A

A


Simravleebi funqciebi mimartebebi simravluri igiveobebi
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 mimartebebi simravluri igiveobebi1
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 mimartebebi simravluri igiveobebi2
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 mimartebebi simravluri igiveobebi3
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 mimartebebi simravluri igiveobebi4
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 mimartebebi simravluri igiveobebi5
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 mimartebebi simravluri igiveobebi venis diagramebis sasualebit
simravleebi, funqciebi, mimarTebebisimravluri igiveobebi venis diagramebis saSualebiT

  • xSirad ufro advilia simravluri igiveobebis gageba venis diagramebis daxatviT.

  • magaliTad ganvixiloT de morganis pirveli kanoni


Simravleebi funqciebi mimartebebi de morganis kanoni
simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:


simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

AB:


Simravleebi funqciebi mimartebebi de morganis kanoni1
simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

AB:


Simravleebi funqciebi mimartebebi de morganis kanoni2
simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:


Simravleebi funqciebi mimartebebi de morganis kanoni3
simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

A:

B:


Simravleebi funqciebi mimartebebi de morganis kanoni4
simravleebi, funqciebi, mimarTebebide morganis kanoni

B:

A:

A:

B: