1. CS 105 Digital Logic Design
Chapter 2
Boolean algebra and logic gates

2. Outline 2.2 Basic Definitions
2.3 Axiomatic Definition of Boolean Algebra.
2.4 Basic Theorems and Properties
2.5 Boolean Functions
2.6 Canonical and Standard Forms
2.7 Other Logic Operations
2.8 Digital Logic Gates

3. 2.2 Basic Definitions (1-2) Boolean Algebra: Closure Associative
Formulated by E.V. Huntington, 1904.
A set of elements B={0,1} and two binary operators + and •
Closure
Associative
x, y  B  x+y  B
x, y  B x•y B
(x+y)+z = x + (y + z)
(x•y)•z = x•(y•z)
Commutative
Identity element
x + y = y + x
x•y = y•x
0+x = x+0 = x
1•x = x•1=x

4. 2.2 Basic Definitions (2-2) Complement (inverse) Distributive Law
 x  B,  x'‚  B (complement of x)
x+x'=1;
x•x'=0
Distributive Law
Over +:
x•(y+z)=(x•y)+(x•z).
Over •:
x+ (y.z)=(x+ y)•(x+z).

5. 2.3 Axiomatic Definition of Boolean Algebra (1-2)
Two- Valued Boolean Algebra: x y
x•y 1 x y
x+y 1
•= AND
+ = OR
‘ = NOT x x’ 1

6. 2.3 Axiomatic Definition of Boolean Algebra (2-2)
Two- Valued Boolean Algebra:
Distributive law: x•(y+z)=(x•y)+(x•z)

7. 2.4 Basic Theorems and Properties (1-7)
Duality Principle
Using Huntington rules, one part may be obtained from the other if the binary operators and the identity elements are interchanged.
Duality done by:
1- Interchange OR and AND Operations.
2- Replace the identity elements 1’s by 0’s and 0’s by 1’s.

8. 2.4 Basic Theorems and Properties (2-7)
Table 2.1 Postulates and Theorems of Boolean algebra
Postulates 2
(a) X
+ 0 = + 1
X + X
(X ‘)’= X
(b) X
. 1 =
. 0
X . X
X . X’
Postulates 5
X + X’ = 1
Theorem 1
Theorem 2
Theorem 3 , involution
Postulate 3,commutative
Theorem 4 , associative
Postulate 4 , distributive
Theorem 5 , deMorgan’s
(a)
X + Y
Y + X =
(X + Y) Z +
X + (Y Z)
X(Y + XY XZ
(X + Y)’
X . Y
(b) XY YX =
(XY) Z
X(Y Z) X
+ YZ
(X + Y)
(X + Z)
(X . Y)’
X’ + Y’
Theorem 6 ,absorption
(a)
X + XY = X
(b)
X(X+Y) = X
Parentheses.
NOT.
AND.
OR.
Operator Precedence

9. 2.4 Basic Theorems and Properties (3-7)

10. 2.4 Basic Theorems and Properties (4-7)

11. 2.4 Basic Theorems and Properties (5-7)

12. Theorem 6(a) Absorption
2.4 Basic Theorems and Properties (6-7)
Truth Table
A table of all possible combinations of x and y variables showing the relation between the variable values and the result of the operation.
Theorem 6(a) Absorption
The theorems of Boolean algebra can be proven by means of truth table.

13. Theorem 5: DeMorgan theorem, (x + y)’ = x’y’
2.4 Basic Theorems and Properties (7-7)
Truth Table
Theorem 5: DeMorgan theorem, (x + y)’ = x’y’

14. Logic Circuit  Boolean Function
2.5 Boolean Functions (1-11)
Logic Circuit  Boolean Function
Example
List the truth table for the follwoing functions and draw their implementation gates.
F1= x + (y’z).
F2= x‘y’z + x’yz + xy‘

15. 2.5 Boolean Functions (1-11)
F1= x + (y’z) X Y Z Y’
Y’*Z
X + (y’ z) 1

16. 2.5 Boolean Functions (1-11)
Truth table

17. Gate Implementation of F1
2.5 Boolean Functions (2-11)
Gate Implementation of F1
F1= x + (y’z).

18. Gate Implementation of F2
2.5 Boolean Functions (3-11)
Gate Implementation of F2
F2= x‘y’z + x’yz + xy‘

19. Gate Implementation F2 after simplifying it
2.5 Boolean Functions (4-11)
Gate Implementation F2 after simplifying it
F2= x‘y’z + x’yz + xy’.
F2= x‘(y’z + yz) + xy’.
F2= x‘(z (y’+ y)) + xy’.
F2= x‘(z (1)) + xy’.
F2= x‘z + xy’.

20. Algebraic Manipulation - Simplification
2.5 Boolean Functions (5-11)
Algebraic Manipulation - Simplification
Example
Simplify the following Boolean functions to a minimum number of literals:
x(x’+y)
=xx’ + xy
=0+xy = xy
x+x’y
=(x+x’)(x+y)
=1(x+y)
= x+y

21. 2.5 Boolean Functions (6-11) DeMorgan‘s Theorem xy + x’z + yz
(x+y)(x+y’)
=x+ xy + xy’+yy’
=x (1+ y + y’) =x
(x+y)(x’+z)(y+z)
=(x+y)(x’+z)
by duality function 4
xy + x’z + yz
= xy + x’z + yz(x+x’)
= xy + x’z + xyz + x’yz
=xy(1+z) + x’z (1+y)
= xy + x’z

22. Complement of a Function Finding the complement of a funtion F
2.5 Boolean Functions (7-11)
Complement of a Function
Complement of a variable x
is x’ (0  1 and 1 0).
The complement of a function F
is x’ and is obtained from an interchange of 0’s for 1’s and 1’s for 0’s in the value of F
The dual of a function
is obtained from the interchange of AND and OR operators and 1’s and 0’s.
Finding the complement of a funtion F
Applying DeMorgan’s theorem as many times as necessary.
Complementing each literal of the dual of F

23. 2-variable DeMorgan’s Theorem 3-variable DeMorgan’s Theorem
2.5 Boolean Functions (8-11)
DeMorgan‘s Theorem
2-variable DeMorgan’s Theorem
(x + y)’ = x’y’.
(xy)’ = x’ + y’.
3-variable DeMorgan’s Theorem

24. Generalized DeMorgan’s Theorem
2.5 Boolean Functions (9-11)
DeMorgan‘s Theorem
Generalized DeMorgan’s Theorem

25. 2.5 Boolean Functions (10-11) Example
A simpler procedure for deriving the complement of a function is to take the dual of the function and complement each literal.
This method follows from the Generalized DeMorgan’s Theorem

26. 2.5 Boolean Functions (11-11)
Example

27. Logic circuit  Boolean function  Truth table
2.6 Canonical and Standard Forms (1-23)
Logic circuit  Boolean function  Truth table
Any Boolean function can be expressed as a sum of minterms.
Any Boolean function can be expressed as a product of maxterms

28. Minterm (or standard products):
2.6 Canonical and Standard Forms (2-23)
Minterms and Maxterms
Minterm (or standard products):
Any binary variable may appear either in its normal form (x) or in its complement form (x').
n variables combined with AND.
n variables can be combined to form 2n minterms.
two Variables: x’y’, x’y, xy’, and xy
A variable of a minterm is:
Prime: if the corresponding bit of the binary number is a 0.
Unprime: if the corresponding bit of the binary number is a 1.
001 => x’y’z (m1)
100 => xy’z’ (m4)
111 => xyz (m7)

29. Maxterm (or standard sums):
2.6 Canonical and Standard Forms (3-23)
Minterms and Maxterms
Maxterm (or standard sums):
n variables combined with OR.
A variable of a maxterm is:
Prime: if the corresponding bit of the binary number is a 1.
Unprime: if the corresponding bit of the binary number is a 0.
Each maxterm is the complement of its corresponding minterm and vice versa.
001 => x + y + z’ (M1)
100 => x’ + y + z (M4)
111 => x’ + y’ + z’ (M7)

30. 2.6 Canonical and Standard Forms (4-23)
Minterms and Maxterms

31. 2.6 Canonical and Standard Forms (5-23)
Expressing Truth Table in Boolean Function
Any Boolean function can be expressed as:
Sum of minterms.
Product of maxterms.
(either 0 or 1 for each term)
Said to be in a canonical form.
Expressing Truth Table in Boolean Function
Forming a minterm for each combination of the variables that produces a one in the function.
Taking the OR of all those terms.
n variables:
22n possible functions.

32. Expressing Truth Table in Boolean Function
2.6 Canonical and Standard Forms (6-23)
Expressing Truth Table in Boolean Function
Example:
Minterms

33. Expressing Truth Table in Boolean Function
2.6 Canonical and Standard Forms (7-23)
Expressing Truth Table in Boolean Function
Maxterms 1 2

34. 2.6 Canonical and Standard Forms (8-23)
Expressing Boolean Function in Sum of Minterms (Method 1 - Supplementing)
Rule
If the function is not in Minterm form
Expanding the expression into a sum of AND terms.
Each term is then inspected to see if it contains all the variables.
If it misses one or more variables, it is ANDed with an expression such as x + x', where x is one of the missing variables.

35. Express the Boolean function F = A + B'C as a sum of minterm.
2.6 Canonical and Standard Forms (9-23)
Expressing Boolean Function in Sum of Minterms (Method 1 - Supplementing)
Example
Express the Boolean function F = A + B'C as a sum of minterm.

36. 2.6 Canonical and Standard Forms (10-23)
Expressing Boolean Function in Sum of Minterms (Method 1 - Supplementing)
Example
Short Notation:
F(A, B, C ) = ∑(1,4,5,6,7)

37. 2.6 Canonical and Standard Forms (11-23)
Expressing Boolean Function in Sum of Minterms (Method 1 - Supplementing)
Short Notation:
F(A, B, C ) = ∑(1,4,5,6,7)
Stands for the ORing of terms
The minterms of the function
List of the variables in
the order taken when the minterm is converted to an AND term

38. 2.6 Canonical and Standard Forms (12-23)
Expressing Boolean Function in Sum of Minterms (Method 2 – Truth Table)
Example
Express the Boolean function F = A + B'C as a sum of minterm, using the truth table
F(A, B, C)
= (1, 4, 5, 6,7)
= (0, 2, 3)
F’(A, B, C)
= (0, 2, 3)
= (1, 4, 5, 6, 7)

39. Expressing Boolean Function in Product of Maxterms
2.6 Canonical and Standard Forms (13-23)
Expressing Boolean Function in Product of Maxterms
Rule
If the function is not in Maxterm form
Use distributive law e.g. x + yz = (x + y)(x + z)
Each term is then inspected to see if it contains all the variables.
If it misses one or more variables, it is ORed with an expression such as xx', where x is one of the missing variables.

40. Expressing Boolean Function in Product of Maxterms
2.6 Canonical and Standard Forms (14-23)
Expressing Boolean Function in Product of Maxterms
Example
Express the Boolean function F = xy + x‘z as a product of maxterms.

41. Expressing Boolean Function in Product of Maxterms
2.6 Canonical and Standard Forms (15-23)
Expressing Boolean Function in Product of Maxterms
Example
Short Notation:
F(A, B, C ) = (0,2,4,5)
The minterms of the function
Stands for the ANDing of terms
List of the variables in
the order taken when the maxterm is converted to an OR term

42. Conversion Between Canonical Forms Canonical conversion procedure
2.6 Canonical and Standard Forms (16-23)
Conversion Between Canonical Forms
Canonical conversion procedure
Consider: F(A, B, C) = ∑(1, 4, 5, 6, 7).
F‘: complement of F = F’(A, B, C) = ∑(0, 2, 3) =m0 +m2 + m3 (Missing minterms)
Compute complement of F’ by DeMorgan’s Theorem:
F = (F‘)‘ = (m0 +m2 + m3)‘ = (m0‘. m2‘. m3‘) = M0 . M2 . M3 = (0, 2,3). (Obtain F in different form)
mj‘=Mj.
Summary
Interchange the symbols ∑ and .
List numbers missing from the original form.
E.g. ∑(1, 4, 5, 6, 7) = (0, 2,3).
Shown by truth table (Table 2-5)

43. Two Canonical Forms of Boolean Algebra from Truth Table
2.6 Canonical and Standard Forms (17-23)
Two Canonical Forms of Boolean Algebra from Truth Table
Example:
Boolean exprexsion: F(x, y, z) = xy + x’z
Truth Table
Canonical Forms
F(x, y, z)
=(1, 3, 6, 7)
=(0, 2, 4, 5)

44. Canonical forms  Standard forms
2.6 Canonical and Standard Forms (18-23)
Standard Forms
Canonical Forms: each minterm or maxterm must contain all the variables. Can be obtained from the truth table.
Standard Forms: the terms that form the function may contain one, two, or any number of literals (variables).
Canonical forms  Standard forms
Sum of Minterms, Product of Maxterms.
Sum of Products, Product of Sums.

45. Two types of standard forms (2-level)
2.6 Canonical and Standard Forms (19-23)
Two types of standard forms (2-level)
Sum of Products
Boolean expression containing AND terms, called product terms with one or more literals each.
Sum denotes ORing of these term.
E.g. F1 = y‘ + xy + x‘yz‘.
Product of Sums
Boolean expression containing OR terms, called sum terms with one or more literals each.
Product denotes ANDing of these term.
E.g. F2 = x (y‘ + z)(x‘ + y + z‘)

46. Standard Form and Logic Circuit Two – level Implementation
2.6 Canonical and Standard Forms (20-23)
Standard Form and Logic Circuit
Two – level Implementation
F1 = y‘ + xy + x‘yz‘

47. Standard Form and Logic Circuit Two– level Implementation
2.6 Canonical and Standard Forms (21-23)
Standard Form and Logic Circuit
Two– level Implementation
F2 = x (y‘ + z)(x‘ + y + z‘)

48. 2.6 Canonical and Standard Forms (22-23)
NonStandard Form and Logic Circuit
Can be changed to a standard form by using distributive law
Three – level Implementation
NonStandard Form:
F3 = AB + C(D + E)

49. 2.6 Canonical and Standard Forms (23-23)
NonStandard Form and Logic Circuit
A two-level implementation is preferred:
produces the least amount of delay through the gates when the signal propagates from the inputs to the output
Two – level Implementation
Standard Form:
F4 = AB + CD + CE

50. Subdivided into Three Categories
2.7 Other Logic Operations (1-3)
There are 22n functions for n binary variables.
For n =2:
Where are 16 possible functions.
AND and OR operators are two of them: x . y and x + y.
Subdivided into Three Categories
Two functions that produce a constant 0 or I.
Four functions with unary operation: complement and transfer.
Ten functions with binary operators that define eight different operations: AND, OR NAND, NOR, exclusive-OR, equivalence (XNOR), inhibition, and implication.

51. Possible functions for the two n variables = 22n
2.7 Other Logic Operations (2-3)
Possible functions for the two n variables = 22n

52. 2.7 Other Logic Operations (3-3)

53. 2.8 Digital Logic Gates (1-9)
Standard Gates
Eight gates are:
AND.
OR.
Complement (NOT) (Inverter = Buffer + Bubble)
Transfer (Buffer)
NAND.
NOR.
XOR.
Equivelance.
Each gate has one or two binary input variables, designated by x and y, and one binary output variable designated by F.

54. 2.8 Digital Logic Gates (2-9)

55. 2.8 Digital Logic Gates (3-9)

56. 2.8 Digital Logic Gates (4-9)
Mulitple Inputs
Inverter and buffere accepts ONLY one variable.
AND and OR functions are Communicative and Associative.

57. 2.8 Digital Logic Gates (5-9)
Mulitple Inputs
NAND and NOR functions are Communicative but not Associative.
Solution
Define multiple NOR (or NAND) gate as a complemented OR (or AND) gate (Section 3-6)

58. 2.8 Digital Logic Gates (6-9)
Mulitple Inputs

59. 2.8 Digital Logic Gates (7-9)

60. 2.8 Digital Logic Gates (8-9)
Mulitple Inputs
XOR and equivalence gates are both Communicative and Associative.
Uncommon, usually constructed with other types of gates.
XOR is an odd function (Section 3.9).

61. 2.8 Digital Logic Gates (9-9)
Mulitple Inputs