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Logika da diskretuli maTematika Tbilisis saxelmwifo universiteti zusti da sabunebismetyvelo mecnierebaTa fakulteti informatikis mimarTuleba bakalavriati.

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Presentation on theme: "Logika da diskretuli maTematika Tbilisis saxelmwifo universiteti zusti da sabunebismetyvelo mecnierebaTa fakulteti informatikis mimarTuleba bakalavriati."— Presentation transcript:

1 logika da diskretuli maTematika Tbilisis saxelmwifo universiteti zusti da sabunebismetyvelo mecnierebaTa fakulteti informatikis mimarTuleba bakalavriati prof. revaz grigolia

2 simravleebi, funqciebi, mimarTebebi leqcia # simravluri operaciebi gaerTianeba da TanaukveTi gaerTianeba –TanakveTa –sxvaoba –damateba –simetriuli sxvaoba

3 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.

4 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.

5 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.

6 simravleebi, funqciebi, mimarTebebi leqcia # 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.

7 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 | } = ?

8 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 } = N pas6: U = R. { |x | } = arauaryofiTi namdvili ricxvebi.

9 simravleebi, funqciebi, mimarTebebi leqcia # 1 Semdegi Teoriul-simravluri operaciebi –gaerTianeba ( ) –TanakveTa ( ) –sxvaoba ( –damateba () –simetriuli sxvaoba ( ) gvaZleven simravleebs: A B, A B, A-B, A B, and A.

10 simravleebi, funqciebi, mimarTebebi leqcia # 1 gaerTianeba elementebi ekuTvnis ori simravlidan erTs mainc A B = { x | x A x B } AB U A B

11 simravleebi, funqciebi, mimarTebebi elementebi ekuTvnis zustad orive simravles TanakveTa A B = { x | x A x B } AB U A B

12 simravleebi, funqciebi, mimarTebebi TanaukveTi simravleebi gans: Tu A da B-s ar gaaCniaT saerTo elementebi, maSin mas uwodeben TanaukveT simravleebs, e. i. A B =. AB U

13 simravleebi, funqciebi, mimarTebebi TanaukveTi gaerTianeba AB U

14 simravleebi, funqciebi, mimarTebebi sxvaoba elementebi ekuTvnis pirvel simravles da ara meores A B = { x | x A x B } A B U A B

15 simravleebi, funqciebi, mimarTebebi simetriuli sxvaoba elementebi ekuTvnis oridan mxolod erT simravles AB UA B A B = { x | x A x B }

16 simravleebi, funqciebi, mimarTebebi damateba elementebi ar ekuTvnis simravles (unaruli operatoria) A = { x | x A } A U A

17 simravleebi, funqciebi, mimarTebebi simravluri igiveobebi logikuri igiveobebi hqmnian simravlur igiveobebs sxvadasxva simravluri operaciebis gansazRvris gamoyenebiT. magaliTad: lema. (gaerTianebis asociaciuroba). (A B ) C = A (B C )

18 simravleebi, funqciebi, mimarTebebi simravluri 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.)

19 simravleebi, funqciebi, mimarTebebi simravluri 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.)

20 simravleebi, funqciebi, mimarTebebi simravluri 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.)

21 simravleebi, funqciebi, mimarTebebi simravluri 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.)

22 simravleebi, funqciebi, mimarTebebi simravluri 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.

23 simravleebi, funqciebi, mimarTebebi simravluri igiveobebi venis diagramebis saSualebiT xSirad ufro advilia simravluri igiveobebis gageba venis diagramebis daxatviT. magaliTad ganvixiloT de morganis pirveli kanoni

24 simravleebi, funqciebi, mimarTebebi de morganis kanoni A:A: B:B:

25 A:A: B:B: A B :

26 simravleebi, funqciebi, mimarTebebi de morganis kanoni A:A: B:B: A B :

27 simravleebi, funqciebi, mimarTebebi de morganis kanoni A:A: B:B:

28 A:A: B:B: A: B:

29 simravleebi, funqciebi, mimarTebebi de morganis kanoni A:A: B:B: A: B:

30 simravleebi, funqciebi, mimarTebebi de morganis kanoni


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