1. COMS 361 Computer Organization
Title: Boolean Algebra
Date: 11/02/2004
Lecture Number: 17

2. Announcements
Homework 8
Due Thursday, 11/11/04

3. Review Representing numbers in binary Problems One’s complement
Unsigned (magnitude)
Sign-magnitude
One’s complement
Two’s complement

4. Outline Representing numbers in binary Boolean Algebra Boolean Logic
Finish overflow
Boolean Algebra
Boolean Logic

5. Truth Table
Provides a listing of every possible combination of inputs and corresponding outputs
Example (2 inputs, 2 outputs):

6. Truth Table
Example (3 inputs, 2 outputs):

7. Proof using Truth Table
Can use truth table to prove a Boolean expression
Prove that: x . (y + z) = (x . y) + (x . z)
(i) Construct truth table for LHS & RHS of above equality

8. Proof using Truth Table
(ii) Check that LHS = RHS
Postulate is SATISFIED because output column 2 & 5 (for LHS & RHS expressions) are equal for all cases

9. Duality
Duality Principle – every valid Boolean expression (equality) remains valid if the operators and identity elements are interchanged, as follows:
+  .
1  0
Example: Given the expression
a + (b.c) = (a+b).(a+c)
then its dual expression is
a . (b+c) = (a.b) + (a.c)

10. Duality Duality gives free theorems
“two for the price of one”
You prove one theorem and the other comes for free!
If (x+y+z)' = x'.y.'z' is valid, then its dual is also valid:
(x.y.z)' = x'+y'+z’
If x + 1 = 1 is valid, then its dual is also valid:
x . 0 = 0

11. Basic Theorems of Boolean Algebra
Apart from the axioms/postulates, there are other useful theorems.
Idempotency
(a) x + x = x (b) x . x = x
Proof of (a):
x + x = (x + x) (identity)
= (x + x).(x + x') (complementarity)
= x + x.x' (distributivity)
= x (complementarity)
= x (identity)

12. Basic Theorems of Boolean Algebra
Theorems can be proved using the truth table method.
They can also be proved by algebraic manipulation using axioms/postulates or other basic theorems.

13. Basic Theorems of Boolean Algebra
absorption can be proved by:
x + x.y = x.1 + x.y (identity)
= x.(1 + y) (distributivity)
= x.(y + 1) (commutativity)
= x (Theorem 2a)
= x (identity)
By duality:
x.(x+y) = x
Try to prove other theorems by algebraic manipulation.

14. Boolean Functions
Boolean function is an expression formed with binary variables, the two binary operators, OR and AND, and the unary operator, NOT, parenthesis and the equal sign.
Its result is also a binary value.
We usually use . for AND, + for OR, and ' or ¬ for NOT. Sometimes, we may omit the . if there is no ambiguity.

15. Boolean Functions From the truth table, F3=F4.
Examples:
F1= xyz'
F2= x + y'z
F3=(x'y'z)+(x'yz)+(xy')
F4=xy'+x'z
From the truth table, F3=F4.
Can you also prove by algebraic manipulation that F3=F4?

16. Complement of Functions
Given a function, F, the complement of this function, F', is obtained by interchanging 1 with 0 in the function’s output values.
Example: F1 = xyz'
Complement:
F1' = (xyz')'
= x' + y' + (z')' DeMorgan
= x' + y' + z Involution

17. Complement of Functions
More general DeMorgan’s theorems useful for obtaining complement functions:
(A + B + C Z)' = A' . B' . C' … . Z'
(A . B . C Z)' = A' + B' + C' + … + Z'

18. Standard Forms
Certain types of Boolean expressions lead to gating networks which are desirable from implementation viewpoint
Two Standard Forms: Sum-of-Products and Product-of-Sums
Literals: a variable on its own or in its complemented form. Examples: x, x' , y, y'
Product Term: a single literal or a logical product (AND) of several literals.
Examples: x, xyz', A'B, AB

19. Standard Forms
Sum Term: a single literal or a logical sum (OR) of several literals.
Examples: x, x+y+z', A'+B, A+B
Sum-of-Products (SOP) Expression: a product term or a logical sum (OR) of several product terms.
Examples: x, x+yz', xy'+x'yz, AB+A'B'
Product-of-Sums (POS) Expression: a sum term or a logical product (AND) of several sum terms.
Exampes: x, x(y+z'), (x+y')(x'+y+z), (A+B)(A'+B')

20. Standard Forms
Every boolean expression can either be expressed as sum-of-products or product-of-sums expression

21. Summary Boolean algebra is required for constructing digital circuits
AND, OR and NOT constitute connectives
There are certain postulates and theorems in Boolean algebra which helps in manipulation of boolean expression
Boolean laws can be proved with either the help of truth table or with the basic postulates
SOP and POS are two generic way of expressing any boolean function

22. Example: Half Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: half adder
adds two binary digits
to form Sum and Carry

23. Example: Half Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: half adder
adds two binary digits
to form Sum and Carry

24. Example: Half Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: half adder
adds two binary digits
to form Sum and Carry

25. Example: Half Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: half adder
adds two binary digits
to form Sum and Carry

26. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

27. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

28. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

29. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

30. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

31. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

32. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

33. Example: Full Adder Truth Tables
Tabulate all possible input combinations and their associated output values
Example: full adder
adds two binary digits and
Carry in to form Sum and
Carry Out
Example: half adder
adds two binary digits
to form Sum and Carry

34. Example: Full Adder A 1 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin

35. Example: Full Adder A 1 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin

36. Example: Full Adder A 1 B 1 Cin 1 Sum 1 Cout 11 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin + A B Cin

37. Example: Full Adder A 1 B 1 Cin 1 Sum 1 Cout 11 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin + A B Cin + A B Cin

38. Example: Full Adder A 1 B 1 Cin 1 Sum 1 Cout 11 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin + A B Cin + A B Cin
Cout = A B Cin

39. Example: Full Adder A 1 B 1 Cin 1 Sum 1 Cout 11 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin + A B Cin + A B Cin
Cout = A B Cin + A B Cin

40. Example: Full Adder A 1 B 1 Cin 1 Sum 1 Cout 11 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin + A B Cin + A B Cin
Cout = A B Cin + A B Cin + A B Cin

41. Example: Full Adder A 1 B 1 Cin 1 Sum 1 Cout 11 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin + A B Cin + A B Cin
Cout = A B Cin + A B Cin + A B Cin + A B Cin

42. Example: Full Adder A 1 B 1 Cin 1 Sum 1 Cout 11 B 1
Cin 1
Sum 1
Cout 1
Sum = A B Cin + A B Cin + A B Cin + A B Cin
Cout = A B Cin + A B Cin + A B Cin + A B Cin

43. Example Circuit to drive a digital display
Need 4 bits to represent 10 numbers (0-9) L1
L 6 L2 L3
L 7
L 4
L 5

44. Example Determine which segments are on for each number L1 L 4 L 6 L2

45. Example Determine which segments are on for each number L1 L 4 L 6 L2

46. Example Determine which segments are on for each number L1 L 4 L 6 L2

47. Example Determine which segments are on for each number L1 L 4 L 6 L2

48. Example Determine which segments are on for each number L1 L 4 L 6 L2

49. Example Determine which segments are on for each number L1 L 4 L 6 L2

50. Example Determine which segments are on for each number L1 L 4 L 6 L2

51. Example Determine which segments are on for each number L1 L 4 L 6 L2

52. Example Determine which segments are on for each number L1 L 4 L 6 L2

53. Example Determine which segments are on for each number L1 L 4 L 6 L2

54. Example
Circuit to drive segment L4 L1
L 6 L2 L3
L 7
L 4
L 5

55. Reducing Complexity
Laws of Boolean algebra can be applied to full adder's carry out
Function to derive the following simplified expression
Cout = A Cin + B Cin + A B

56. Some standard gates and their symbols and truth tables