1. Boolean Algebra and Logic Gates
Chapter 2
Boolean Algebra and
Logic Gates

2. Chapter 2. Boolean Algebra and Logic Gates
2-1 Introduction
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 2

3. 2-3 Axiomatic Definition of Boolean Algebra
2-2 Basic Definitions
2-3 Axiomatic Definition of Boolean Algebra
• Boolean Algebra
forrmulated by E.V. Huntington, 1904)
A set of elements B={0,1} and tow binary operators + and•
1. Closure
x, y  B  x+y B; x, y  B  x•y B
2. Associative
(x+y)+z = x + (y + z); (x•y)•z = x • (y•z)
3. Commutative
x+y = y+x; x•y = y•x
4. an identity element
0+x = x+0 = x; •x = x•1=x  e,x B
  x  B,  x'  B (complement of x)
x+x'=1; x•x'=0 (o,1 - identity elements w.r.t. +, .
6. distributive Law over + : x•(y+z)=(x•y)+(x•z)
distributive over x: x+ (y.z)=(x+ y)•(x+ z) 3

4. Two-valued Boolean Algebra
•= AND
+ = OR
‘ = NOT
Distributive law: x•(y+z)=(x•y)+(x•z) 4

5. 2-4 Basic Theorems and Properties
Duality Principle:
Using Huntington rules, one part may be obtained from the other if the binary operators and the identity elements are interchanged 5

6. 2-4 Basic Theorems and Properties
Operator Precedence
1. parentheses
2. NOT
3. AND
4. OR 5

7. Basic Theorems 6

8. 7

9. Verification by 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
Theorem 5. DeMorgan 8

10. 2-5 Boolean Functions Logic Circuit  Boolean Function
Boolean Fxnctions F
= x + (y’z) F
= x‘y’z + x’yz + xy’
1 2 9

11. Boolean Function F2
F2 = x’y’z + x’yz + xy’ 10

12. Algebraic Manipulation - Simplification
Example 2.1 Simplify the following Boolean functions to a minimum number of literals:
1- x(x’+y)
=xx’ + xy
=0+xy=xy
2- x+x’y
=(x+x’)(x+y)
=1(x+y)
= x+y

13. 3-(x+y)(x+y’)
=x+xy+xy’+yy’
=x (1+ y + y’) =x
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

14. Complement of a Function
DeMorgan’s Theorem
•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 function F
Applying DeMorgan’s theorem as many times as necessary
complementing each literal of the dual of F 13

15. DeMorgan’s Theorem 2-variable DeMorgan’s Theorem
(x + y)’ = x’y’ and (xy)’ = x’ + y’
3-variable DeMorgan’s Theorem
Generalized DeMorgan’s Theorem 12

16. H.W: 1,2,3,4,6,8,9 14

17. 2-5 Canonical and Standard Forms
• Minterms and maxterms
– Expressing combinationsof 0’s and 1’s with binary variables
• Logic circuit  Boolean function  Truth table
– Any Boolean function can be expressed as a sum of minterms
- Any Boolean functiox can be expressed as a product of
maxterms
• Canonical and Standard Forms 15

18. Minterms and Maxterms
Minterm (or standard product): Maxterm (or standard sum):
– n variables combined with AND – n variables combined with OR
– n variables can be combined to – A variable of a maxterm is
form 2 minterms n
• unprimed is the corresponding
• two Variables: x’y’, x’y, xy’, and xy
bit is a 0
– A variable of a minterm is
• and primed if a 1
• primed if the corresponding bit of
the binary number is a 0,
001 => x’y’z
• and unprimed if a 1
100 => xy’z’ 16
111 => xyz

19. Expressing Truth Table in Boolean Function
• Any Boolean function
can be expressed
a sum of minterms or
a product of maxterms
(either 0 or 1 for each term)
• said to be in a canonical
form
• x variables
 2 minterms n
 2 possible functions 2n 17

20. Expressing Boolean Function in Sum of
Minterms (Method 1 - Supplementing) 18

21. Expressing Boolean Function in Sum of
Minterms (method 2 – 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) 19

22. Expressing Boolean Function in Product of Maxterms

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) = m
+ m
+ m
0 2 3
Compute complement of F’ by DeMorgan’s Theorem
F = (F’)’ = (m
+ m
+ m
)‘ = (m
’  m
’  m ’)
= m
’ m
’ m
’ = M M M
(0, 2, 3) 
 Summary •
m ’ = M
j j
• Conversion between product of maxterms and sum of minterms 
(1, 4, 5, 6, 7) = (0, 2, 3)
• Shown by truth table (Table 2-5) 21

24. Example:– Two Canonical Forms of Boolean Algebra from Truth Table
Boolean expression: x(x, y, z) = xy + x’z
Deriving the truth table
Expressing in canonical forms
x(x, y, z) = (1, 3, 6, 7) = (0, 2, 4, 5) x2

25. Standard Forms * Canonical forms: each minterm or maxterm must
contain all the variables
* Standard forms: the terms that form the function
may contain one, two, or any number of literals
(variables)
• Two types of standard forms (2-level)
– sum of products F
= y’ + xy + x’yz’ 1
– Product of sums F
= x(y’ + z)(x’ + y + x’) 2
• Canonical forms  Standard forms
– sum of minterms, Product of maxterms
– Sum of products, Product of sums 23

26. Standard Form and Logic Circuit
= y’ + xy + x’yz’ F
= x(y’ + z)(x’ + y + z) 1 2 24

27. Nonstandard Form and Logic Circuit
Nonstandard form: Standard form: F
= AB + C(D+E) F
= AB + CD + CE
3 3
A two-level implementation is preferred: produces the least amount of delas
Through the gates when the signal propagates from the inputs to the output 25

28. 2-7 Other Logic Operations
• There are 2 function 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: 26

29. Truth Tables and Boolean Expressions for
the 16 Functions of Two Variables 2x

30. 2-8 Digital Logic
Gates 28
Figure 2-5 Digital Logic Gates

31. Extension to Multiple-Inputs
• NAND and NOR functions are
communicative but not Associative
– Define multiple NOR (or NAND) gate as a
complemented OR (or AND) gate (Section 3-6)
XOR and equivalence gates are both
communicative and associative
– uncommon, usually constructed with other gates
– XOR is an odd function (Section 3-8) 29

32. 30

33. Positive and Negative logic
Logic value
Logic value
Signal value
Signal value H H 1 L L 1
(a) Positive logic
(b) Negative logic