1. Axiomas Teoremas Exemplos
´
Algebra Booleana
Prof. Ricardo Menotti
Universidade Federal de S˜o Carlos (UFSCar)
a
menotti@dc.ufscar.br
Prof. Ricardo Menotti ´
Algebra Booleana

2. Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0 1b 1+1=1
Prof. Ricardo Menotti ´
Algebra Booleana

3. Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0 1b 1+1=1
2a 1.1 = 1 2b 0+0=0
Prof. Ricardo Menotti ´
Algebra Booleana

4. Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0 1b 1+1=1
2a 1.1 = 1 2b 0+0=0
3a 0.1 = 1.0 = 0 3b 1+0=0+1=1
Prof. Ricardo Menotti ´
Algebra Booleana

5. Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0 1b 1+1=1
2a 1.1 = 1 2b 0+0=0
3a 0.1 = 1.0 = 0 3b 1+0=0+1=1
4a Se x = 0, ent˜o x = 1
a 4b Se x = 1, ent˜o x = 0
a
Prof. Ricardo Menotti ´
Algebra Booleana

6. Axiomas Teoremas Exemplos
Teoremas com uma vari´vel
a
5a x.0 = 0 5b x +1=1
Prof. Ricardo Menotti ´
Algebra Booleana

7. Axiomas Teoremas Exemplos
Teoremas com uma vari´vel
a
5a x.0 = 0 5b x +1=1
6a x.1 = x 6b x +0=x
Prof. Ricardo Menotti ´
Algebra Booleana

8. Axiomas Teoremas Exemplos
Teoremas com uma vari´vel
a
5a x.0 = 0 5b x +1=1
6a x.1 = x 6b x +0=x
7a x.x = x 7b x +x =x
Prof. Ricardo Menotti ´
Algebra Booleana

9. Axiomas Teoremas Exemplos
Teoremas com uma vari´vel
a
5a x.0 = 0 5b x +1=1
6a x.1 = x 6b x +0=x
7a x.x = x 7b x +x =x
8a x.x = 0 8b x +x =1
Prof. Ricardo Menotti ´
Algebra Booleana

10. Axiomas Teoremas Exemplos
Teoremas com uma vari´vel
a
5a x.0 = 0 5b x +1=1
6a x.1 = x 6b x +0=x
7a x.x = x 7b x +x =x
8a x.x = 0 8b x +x =1
9 x =x
Prof. Ricardo Menotti ´
Algebra Booleana

11. Axiomas Teoremas Exemplos
Teoremas com duas ou mais vari´veis
a
Comutativas
10a x.y = y .x 10b x +y =y +x
Prof. Ricardo Menotti ´
Algebra Booleana

12. Axiomas Teoremas Exemplos
Teoremas com duas ou mais vari´veis
a
Comutativas
10a x.y = y .x 10b x +y =y +x
Associativas
11a x.(y .z) = (x.y ).z 11b x + (y + z) = (x + y ) + z
Prof. Ricardo Menotti ´
Algebra Booleana

13. Axiomas Teoremas Exemplos
Teoremas com duas ou mais vari´veis
a
Comutativas
10a x.y = y .x 10b x +y =y +x
Associativas
11a x.(y .z) = (x.y ).z 11b x + (y + z) = (x + y ) + z
Distributivas
12a x.(y + z) = x.y + x.z 12b x + y .z = (x + y ).(x + z)
Prof. Ricardo Menotti ´
Algebra Booleana

14. Axiomas Teoremas Exemplos
Teoremas com duas ou mais vari´veis
a
Absor¸˜o
ca
13a x + x.y = x 13b x.(x + y ) = x
Prof. Ricardo Menotti ´
Algebra Booleana

15. Axiomas Teoremas Exemplos
Teoremas com duas ou mais vari´veis
a
Absor¸˜o
ca
13a x + x.y = x 13b x.(x + y ) = x
Combina¸˜o
ca
14a x.y + x.y = x 14b (x + y ).(x + y ) = x
Prof. Ricardo Menotti ´
Algebra Booleana

16. Axiomas Teoremas Exemplos
Teoremas com duas ou mais vari´veis
a
Absor¸˜o
ca
13a x + x.y = x 13b x.(x + y ) = x
Combina¸˜o
ca
14a x.y + x.y = x 14b (x + y ).(x + y ) = x
DeMorgan
15a x.y = x + y 15b x + y = x.y
16a x + x.y = x + y 16b x.(x + y ) = x.y
Prof. Ricardo Menotti ´
Algebra Booleana

17. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
Prof. Ricardo Menotti ´
Algebra Booleana

18. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
Prof. Ricardo Menotti ´
Algebra Booleana

19. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
Prof. Ricardo Menotti ´
Algebra Booleana

20. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
Prof. Ricardo Menotti ´
Algebra Booleana

21. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
LHS = x1 .x 1 .(x2 + x3 ) + x1 .x 3 .(x2 + x3 ) + x2 .x 1 .(x2 + x3 ) + x2 .x 3 .(x2 + x3 )
Prof. Ricardo Menotti ´
Algebra Booleana

22. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
LHS = x1 .x 1 .(x2 + x3 ) + x1 .x 3 .(x2 + x3 ) + x2 .x 1 .(x2 + x3 ) + x2 .x 3 .(x2 + x3 )
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x 3 .x2 +x1 .x 3 .x3 +x2 .x 1 .x2 +x2 .x 1 .x3 +x2 .x 3 .x2 +x2 .x 3 .x3
Prof. Ricardo Menotti ´
Algebra Booleana

23. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
LHS = x1 .x 1 .(x2 + x3 ) + x1 .x 3 .(x2 + x3 ) + x2 .x 1 .(x2 + x3 ) + x2 .x 3 .(x2 + x3 )
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x 3 .x2 +x1 .x 3 .x3 +x2 .x 1 .x2 +x2 .x 1 .x3 +x2 .x 3 .x2 +x2 .x 3 .x3
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x2 .x 3 +x1 .x3 .x 3 +x 1 .x2 .x2 +x 1 .x2 .x3 +x2 .x2 .x 3 +x2 .x3 .x 3
Prof. Ricardo Menotti ´
Algebra Booleana

24. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
LHS = x1 .x 1 .(x2 + x3 ) + x1 .x 3 .(x2 + x3 ) + x2 .x 1 .(x2 + x3 ) + x2 .x 3 .(x2 + x3 )
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x 3 .x2 +x1 .x 3 .x3 +x2 .x 1 .x2 +x2 .x 1 .x3 +x2 .x 3 .x2 +x2 .x 3 .x3
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x2 .x 3 +x1 .x3 .x 3 +x 1 .x2 .x2 +x 1 .x2 .x3 +x2 .x2 .x 3 +x2 .x3 .x 3
LHS = 0.x2 + 0.x3 + x1 .x2 .x 3 + x1 .0 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3 + x2 .0
Prof. Ricardo Menotti ´
Algebra Booleana

25. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
LHS = x1 .x 1 .(x2 + x3 ) + x1 .x 3 .(x2 + x3 ) + x2 .x 1 .(x2 + x3 ) + x2 .x 3 .(x2 + x3 )
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x 3 .x2 +x1 .x 3 .x3 +x2 .x 1 .x2 +x2 .x 1 .x3 +x2 .x 3 .x2 +x2 .x 3 .x3
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x2 .x 3 +x1 .x3 .x 3 +x 1 .x2 .x2 +x 1 .x2 .x3 +x2 .x2 .x 3 +x2 .x3 .x 3
LHS = 0.x2 + 0.x3 + x1 .x2 .x 3 + x1 .0 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3 + x2 .0
LHS = 0 + 0 + x1 .x2 .x 3 + 0 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3 + 0
Prof. Ricardo Menotti ´
Algebra Booleana

26. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
LHS = x1 .x 1 .(x2 + x3 ) + x1 .x 3 .(x2 + x3 ) + x2 .x 1 .(x2 + x3 ) + x2 .x 3 .(x2 + x3 )
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x 3 .x2 +x1 .x 3 .x3 +x2 .x 1 .x2 +x2 .x 1 .x3 +x2 .x 3 .x2 +x2 .x 3 .x3
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x2 .x 3 +x1 .x3 .x 3 +x 1 .x2 .x2 +x 1 .x2 .x3 +x2 .x2 .x 3 +x2 .x3 .x 3
LHS = 0.x2 + 0.x3 + x1 .x2 .x 3 + x1 .0 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3 + x2 .0
LHS = 0 + 0 + x1 .x2 .x 3 + 0 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3 + 0
LHS = x1 .x2 .x 3 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3
Prof. Ricardo Menotti ´
Algebra Booleana

27. Axiomas Teoremas Exemplos
Provar que:
(x1 + x2 ).(x2 + x3 ).(x 1 + x 3 ) = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
LHS = (x1 + x2 ).(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).(x 1 + x 3 ) + x2 .(x2 + x3 ).(x 1 + x 3 )
LHS = x1 .(x2 + x3 ).x 1 + x1 .(x2 + x3 ).x 3 + x2 .(x2 + x3 ).x 1 + x2 .(x2 + x3 ).x 3
LHS = x1 .x 1 .(x2 + x3 ) + x1 .x 3 .(x2 + x3 ) + x2 .x 1 .(x2 + x3 ) + x2 .x 3 .(x2 + x3 )
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x 3 .x2 +x1 .x 3 .x3 +x2 .x 1 .x2 +x2 .x 1 .x3 +x2 .x 3 .x2 +x2 .x 3 .x3
LHS = x1 .x 1 .x2 +x1 .x 1 .x3 +x1 .x2 .x 3 +x1 .x3 .x 3 +x 1 .x2 .x2 +x 1 .x2 .x3 +x2 .x2 .x 3 +x2 .x3 .x 3
LHS = 0.x2 + 0.x3 + x1 .x2 .x 3 + x1 .0 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3 + x2 .0
LHS = 0 + 0 + x1 .x2 .x 3 + 0 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3 + 0
LHS = x1 .x2 .x 3 + x 1 .x2 .x2 + x 1 .x2 .x3 + x2 .x2 .x 3
LHS = x1 .x2 .x 3 + x 1 .x2 + x 1 .x2 .x3 + x2 .x 3
Prof. Ricardo Menotti ´
Algebra Booleana

28. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
Prof. Ricardo Menotti ´
Algebra Booleana

29. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
Prof. Ricardo Menotti ´
Algebra Booleana

30. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
Prof. Ricardo Menotti ´
Algebra Booleana

31. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
Prof. Ricardo Menotti ´
Algebra Booleana

32. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
Prof. Ricardo Menotti ´
Algebra Booleana

33. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
Prof. Ricardo Menotti ´
Algebra Booleana

34. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
RHS = x.z + x.z + x.y .w
Prof. Ricardo Menotti ´
Algebra Booleana

35. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
RHS = x.z + x.z + x.y .w
RHS = x.z.y + x.z.y + x.z.y + x.z.y + x.y .w
Prof. Ricardo Menotti ´
Algebra Booleana

36. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
RHS = x.z + x.z + x.y .w
RHS = x.z.y + x.z.y + x.z.y + x.z.y + x.y .w
RHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.y .w .z+ x.y .w .z
Prof. Ricardo Menotti ´
Algebra Booleana

37. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
RHS = x.z + x.z + x.y .w
RHS = x.z.y + x.z.y + x.z.y + x.z.y + x.y .w
RHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.y .w .z+ x.y .w .z
RHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
Prof. Ricardo Menotti ´
Algebra Booleana

38. Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y .z.w = x.z + x.z + x.y .w
LHS = x.z + x.z + y .z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y .z.w
LHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + y .z.w .x + y .z.w .x
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
LHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
RHS = x.z + x.z + x.y .w
RHS = x.z.y + x.z.y + x.z.y + x.z.y + x.y .w
RHS = x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.z.y .w + x.y .w .z+ x.y .w .z
RHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
RHS = x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w + x.y .z.w
Prof. Ricardo Menotti ´
Algebra Booleana