attachment:balint_3.txt of DiskretnaMatematikaALogika/Premie

Attachment 'balint_3.txt'

Dobry den,
posielam riesenie premioveho prikladu z dnesnej prednasky:

Dokazte pomocou rovnosti:
1) A+0=A   A*U=A
2) A+U=U   A*0=0
3) !0=U    !U=0
4) !(A+B)=!A*!B

1)
A + 0 = A + (A * !A) = A
A * U = A * (A + !A) = A

2)
A + U = A + (A + !A) = (A + A) + !A = A + !A = U
A * 0 = A * (A * !A) = (A * A) * !A = A * !A = 0

3)
!0 = !0 + 0 = U
!U = !U * U = 0

4)
!(A+B)
= !(A+B) * U * U
= !(A+B) * ((U+A) * (U+B))
= !(A+B) * ((A+B+!B) * (B+A+!A))
= !(A+B) * ((A+B)+!B) * (A+B)+!A))
= !(A+B) * ((A+B) + (!A*!B))
= (!(A+B) * (A+B)) + (!(A+B) * (!A*!B))
= 0 + (!(A+B) * (!A*!B))
= 0 + 0 + (!(A+B) * (!A*!B))
= (0*!A) + (0*!B) + (!(A+B) * (!A*!B))
= (!A*B*!B) + (!B*A*!A) + (!(A+B) * (!A*!B))
= ((!A*!B)*B) + ((!A*!B)*A) + (!(A+B) * (!A*!B))
= ((!A*!B) * (A+B)) + (!(A+B) * (!A*!B))
= (!A+!B) * ((A+B) + !(A+B))
= (!A*!B) * U
= !A*!B


Este ku vsetkym trom rieseniam - aj z predchadzajucich dvoch e-mailov (hlavne
kvoli uplnosti):
A*B - prienik dvoch mnozin
A+B - zjednotenie dvoch mnozin
!A  - komplement (doplnok) mnoziny
0   - prazdna mnozina
U   - univerzum

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[get | view] (2008-04-25 09:01:16, 0.5 KB) [[attachment:balint_1.txt]]
[get | view] (2008-04-25 09:01:16, 1.0 KB) [[attachment:balint_3.txt]]
[get | view] (2008-04-25 09:01:16, 137.9 KB) [[attachment:gyurasz_6.pdf]]
[get | view] (2008-04-25 09:01:16, 25.5 KB) [[attachment:krautschneider_3.doc]]
[get | view] (2008-04-25 09:01:16, 31.5 KB) [[attachment:rybanska_6.doc]]

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