1. Topic1: Boolean Algebra José Nelson Amaral
CMPUT329 - Fall 2003
Topic1: Boolean Algebra
José Nelson Amaral
CMPUT 329 Computer Organization and Architecture II

2. CMPUT 329 Computer Organization and Architecture II
Reading Assignment
Wakerly, Chapter 4
(Sections 4.1.1, 4.1.2, 4.1.3)
CMPUT 329 Computer Organization and Architecture II

3. What is a switching network?X1 Xm X2 Z1 Zm Z2
Combinatorial Network: A stateless network. The output is
completely determined by the values of the input.
Sequential Network: The network stores an internal
state. The output is determined by the input,
and by the internal state.
CMPUT 329 Computer Organization and Architecture II

4. Logic Functions: Boolean Algebra
INVERTER X X’
If X=0 then X’=1
If X=1 then X’=0 A B
C=A·B
If A=1 AND B=1 then C=1
otherwise C=0
AND OR A B
C=A+B
If A=1 OR B=1 then C=1
otherwise C=0
CMPUT 329 Computer Organization and Architecture II

5. Boolean expressions and logic circuits
Any Boolean expression can be implemented as a logic circuit.
X = [A(C+D)]’+BE C D
C+D A
A(C+D)
[A(C+D)]’
[A(C+D)]’+BE B E BE
CMPUT 329 Computer Organization and Architecture II

6. Basic Theorems: Operations with 0 and 1
X+0 = X X
C=X
X+1 = 1 X 1
C=1 X 1
C=X
X·1 = X X
C=0
X·0 = 0
CMPUT 329 Computer Organization and Architecture II

7. Basic Theorems: Idempotent Laws
X+X = X X
C=X X
C=X
X·X = X
CMPUT 329 Computer Organization and Architecture II

8. Basic Theorems: Involution Law
(X’)’=X B X
C=X
CMPUT 329 Computer Organization and Architecture II

9. Basic Theorems: Laws of Complementarity
X+X’ = 1 X X’
C=1 X X’
C=0
X·X’ = 0
CMPUT 329 Computer Organization and Architecture II

10. Expression Simplification using the Basic Theorems
X can be an arbitrarily complex expression.
Simplify the following boolean expressions as
much as you can using the basic theorems.
(AB’ + D)E + 1 = 1
(AB’ + D)(AB’ + D)’ = 0
(AB + CD) + (CD + A) + (AB + CD)’ = 1
(AB’ + D)E + 1 =
(AB’ + D)(AB’ + D)’ =
(AB + CD) + (CD + A) + (AB + CD)’ =
CMPUT 329 Computer Organization and Architecture II

11. CMPUT 329 Computer Organization and Architecture II
Associative Law
(X+Y)+Z = X+(Y+Z) X Y Z C
CMPUT 329 Computer Organization and Architecture II

12. CMPUT 329 Computer Organization and Architecture II
Associative Law
(XY)Z = X(YZ) X Y Z C Y Z X C
CMPUT 329 Computer Organization and Architecture II

13. First Distributive Law
X(Y+Z) = XY+XZ
CMPUT 329 Computer Organization and Architecture II

14. First Distributive Law
X(Y+Z) = XY+XZ
CMPUT 329 Computer Organization and Architecture II

15. First Distributive Law
X(Y+Z) = XY+XZ
CMPUT 329 Computer Organization and Architecture II

16. First Distributive Law
X(Y+Z) = XY+XZ
CMPUT 329 Computer Organization and Architecture II

17. First Distributive Law
X(Y+Z) = XY+XZ
CMPUT 329 Computer Organization and Architecture II

18. Second Distributive Law
X+YZ = (X+Y)(X+Z)
CMPUT 329 Computer Organization and Architecture II

19. Second Distributive Law
X+YZ = (X+Y)(X+Z)
CMPUT 329 Computer Organization and Architecture II

20. Second Distributive Law (A different proof)
(X + Y)(X + Z)
= X(X + Z) + Y(X + Z)
(using the first distributive law)
= XX + XZ + YX + YZ
(using the first distributive law)
= X + XZ + YX + YZ
(using the idempotent law)
= X·1 + XZ + YX + YZ
(using the operation with 1 law)
= X(1 + Z + Y) + YZ
(using the first distributive law)
= X·1 + YZ
(using the operation with 1 law)
= X + YZ
(using the operation with 1 law)
CMPUT 329 Computer Organization and Architecture II

21. Simplification Theorems
XY + XY’ = X
XY + XY’ = X(Y + Y’) = X·1 = X
(X + Y)(X + Y’) = X
(X + Y)(X + Y’) = XX + XY’ + YX + YY’
= X + X(Y’ + Y) + 0
= X + X·1
= X
X + XY = X
X(1 + Y) = X·1 = X
X(X + Y) = X
X(X + Y) = XX + XY = X·1 + XY
= X(1 + Y) = X·1 = X
XY’ + Y = X + Y
(using the second distributive law)
XY’ + Y = Y + XY’ = (Y + X)(Y + Y’)
= (Y + X)·1 = X + Y
(X + Y’)Y = XY
XY + Y’Y = XY + 0 = XY
CMPUT 329 Computer Organization and Architecture II

22. Examples Simplify the following expressions:
W = [M + N’P + (R + ST)’][M + N’P + R + ST]
X = M + N’P Y = R + ST
W = (X + Y’)(X + Y)
W = XX + XY + Y’X + Y’Y
W = X·1 + XY + XY’ + 0
W = X + X(Y + Y’) = X + X·1 = X
W = M + N’P
CMPUT 329 Computer Organization and Architecture II