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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
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