logika da diskretuli maTematika

Tbilisis saxelmwifo universitetizusti da sabunebismetyvelo mecnierebaTa fakulteti

informatikis mimarTulebabakalavriati

prof. revaz grigolia

simravleebi, funqciebi, mimarTebebi leqcia # 1

• 1.1 simravluri operaciebi

• gaerTianeba da TanaukveTi gaerTianeba – TanakveTa – sxvaoba “”– damateba “ ”– simetriuli sxvaoba

simravleebi, funqciebi, mimarTebebi leqcia # 1

universaluri 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 # 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 # 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 # 1

rasselis paradoqsi2. 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 # 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 # 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 } = Npas6: U = R. { |x | } = arauaryofiTi namdvili ricxvebi.

simravleebi, funqciebi, mimarTebebi leqcia # 1

Semdegi Teoriul-simravluri operaciebi

– gaerTianeba ()– TanakveTa ()– sxvaoba (– damateba (“—”)– simetriuli sxvaoba ()

gvaZleven simravleebs: AB, AB, A-B, AB, andA .

simravleebi, funqciebi, mimarTebebi leqcia # 1

gaerTianeba• elementebi ekuTvnis ori simravlidan erTs mainc

AB = { x | x A x B }

A B

U

AB

simravleebi, funqciebi, mimarTebebi

• elementebi ekuTvnis zustad orive simravlesTanakveTa

AB = { x | x A x B }

A B

U

AB

simravleebi, funqciebi, mimarTebebi TanaukveTi simravleebi

• gans: Tu A da B-s ar gaaCniaT saerTo elementebi, maSin mas uwodeben TanaukveT simravleebs, e. i. A B = .

A B

U

simravleebi, funqciebi, mimarTebebi

TanaukveTi gaerTianeba

A B

U

BA

simravleebi, funqciebi, mimarTebebi sxvaoba

• elementebi ekuTvnis pirvel simravles da ara meores

AB = { x | x A x B }

A

B

U

AB

simravleebi, funqciebi, mimarTebebi simetriuli sxvaoba

• elementebi ekuTvnis oridan mxolod erT simravles

A B

UAB

AB = { x | x A x B }

simravleebi, funqciebi, mimarTebebi damateba

• elementebi ar ekuTvnis simravles (unaruli operatoria)

A = { x | x A }

A

U

A

simravleebi, funqciebi, mimarTebebi

simravluri igiveobebi

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

• lema. (gaerTianebis asociaciuroba). (AB )C = A(B C )

simravleebi, funqciebi, mimarTebebi

simravluri igiveobebi

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

• lema. (gaerTianebis asociaciuroba). (AB )C = A(B C )

damtkiceba. (AB )C = {x | x A B x C } (gans. Tan.)

simravleebi, funqciebi, mimarTebebi

simravluri igiveobebi

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

• lema. (gaerTianebis asociaciuroba). (AB )C = A(B C )

damtkiceba. (AB )C = {x | x A B x C } (gans. Tan.)

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

simravleebi, funqciebi, mimarTebebi

simravluri igiveobebi

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

• lema. (gaerTianebis asociaciuroba). (AB )C = A(B C )

damtkiceba. (AB )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 igiveobebi

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

• lema. (gaerTianebis asociaciuroba). (AB )C = A(B C )

damtkiceba. (AB )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 igiveobebi

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

• lema. (gaerTianebis asociaciuroba). (AB )C = A(B C )

damtkiceba. (AB )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

• xSirad ufro advilia simravluri igiveobebis gageba venis diagramebis daxatviT.

• magaliTad ganvixiloT de morganis pirveli kanoni

BABA

simravleebi, funqciebi, mimarTebebi

de morganis kanoni

A: B:

simravleebi, funqciebi, mimarTebebi

de morganis kanoni

A: B:

AB :

simravleebi, funqciebi, mimarTebebi

de morganis kanoni

A: B:

AB :

:BA

simravleebi, funqciebi, mimarTebebi

de morganis kanoni

A: B:

simravleebi, funqciebi, mimarTebebi

de morganis kanoni

A: B:

A: B:

simravleebi, funqciebi, mimarTebebi

de morganis kanoni

A: B:

A: B:

:BA

simravleebi, funqciebi, mimarTebebi

de morganis kanoni

BA

BA