1. CHAPTER 2 Boolean Algebra
This chapter in the book includes:
Objectives
Study Guide
2.1 Introduction
Basic Operation
2.3 Boolean Expression and Truth Table
2.4 Basic Theorem
2.5 Commutative, Associative and Distributive Laws
Simplification Theorem
Multiplying Out and Factoring
DeMorgan’s Laws
Problems

2. Objectives Topics introduced in this chapter:
Understand the basic operations and laws of Boolean algebra
Relate these operations and laws to AND, OR, NOT gates and switches
Prove these laws using a truth table
Manipulation of algebraic expression using
- Multiplying out
- Factoring
- Simplifying
- Finding the complement of an expression

3. 2.1 Introduction
Digital Circuits: being consist of transistors, diodes, resistors, hardware functional elements to process binary (0,1) data.
->Each basic circuit is called as a gate processing a logical operation(AND, OR,NOT,..). Each gate has one or more input ports and one output port (sometime with its complement).
Boolean algebra: mathematical basis to specify the operations of gates. It is used to describe the inter-connections of digital gates and circuit diagrams.
->Switching algebra, two-valued Boolean algebra, or Binarly logic is a special case of Boolean algebra. It has two state values, 0 and 1.

4. George Boole ( )

5. 2.1 Introduction
we are assuming ‘two-valued Boolean algebra’.
Boolean variable or binary variable : A, B, X, Y, … can only have one of two state values (0, 1)
- representing True(1) or False (0)
positive logic: 1 for High, 0 for Low
negative logic: 1 for Low, 0for High
*In TTL, data sheet may say,
High: above 2.7 V, Low: below 0.8 V,
undefined in the middle.

6. 2.2 Basic Operations Boolean Operation, NOT: ‘
Circle indicates inversion.
Gate Symbol: NOT gate, or Inverter

7. 2.2 Basic Operations A B C = A · B 0 0 0 1 1 0 1 1 1 AND: . or nothing
*Note that it is not an
arithmetic but a Boolean
operation.
A B
C = A · B
0 0
0 1
1 0
1 1 1
Truth Table
*Also known as
logical product.
Logic Gate Symbol

8. 2.2 Basic Operations A B C = A + B 0 0 0 1 1 0 1 1 1 OR: + Truth Table
0 0
0 1
1 0
1 1 1
Truth Table
*Also known as
logical summation.
Gate Symbol

10. 2.2 Basic Operations
Apply to Switch
AND OR

11. 2.3 Boolean Expressions and Truth Tables
Boolean Expression : basic operations to binary variables, constants (0, 1), logic operation symbols, ( ..), and/or equal sign (=). It is also used to show input to output relationship of binary variables.
Priority: parentheses -> NOT -> AND -> OR
Circuit of logic gates : being corresponded to each Boolean expression D=

12. 2.3 Boolean Expressions and Truth Tables
Logic Expression :
Circuit of logic gates : Y=
Logic Evaluation : substituting a value to each variable as A=B=C=1, D=E=0
Literal : a variable or its complement in a logic expression
10 literals, 4 terms

13. 2-Input Circuit and Truth Table
2.3 Boolean Expressions and Truth Tables
2-Input Circuit and Truth Table
Truth Table: specifies the output
of the circuit for all possible
combinations of input values.
A B A’
F = A’ + B
0 0
0 1
1 0
1 1 1

14. 2.3 Boolean Expressions and Truth Tables
Proof using Truth Table: two expressions are equal if they have the same output values for every possible input combinations.
n times
n variables need rows
TABLE 2.1
A B C B’
AB’
AB’ + C
A + C
B’ + C
(A + C) (B’ + C) 1

15. 2.4 Basic Theorems Operations with 0, 1 Idempotent Laws
Involution Laws
Complementary Laws
Proof
Example

16. 2.4 Basic Theorems Duality:
The dual of an algebraic expression is obtained by replacing OR to AND, AND to OR, 1 to 0, and 0 to 1.
If there is a valid theorem, then its dual is also a valid theorem.
(* Variables are not changed and operational priority must be kept.)
EX: A + B* C + 0 => A* (B+C)*1

17. 2.5 Commutative, Associative, and Distributive Laws
Some useful laws applied in the Boolean algebra.
Commutative Laws: ordering of variables is not important in the definition of AND and OR operations.
Associative Laws: among the three variables forming the AND or OR operation, the result is independent of which pair of variables is associated first.
So, parentheses can be
omitted. This requires
hardware attentions, too.

18. Associative Laws for AND and OR
Replacement to 3-input gates

19. 2.5 Commutative, Associative, and Distributive Laws
Multiple AND:
Multiple OR:
Distributive Laws:
Valid only Boolean algebra not for ordinary algebra
OR also can be distributed over AND.
Proof

20. 2.6 Simplification Theorems
Useful Theorems for Simplification
Proof
NOTE: the others are valid as the duals, simplifying the expression results in
the reduction of hardware circuits.
The first two theorems are the basics for further simplification methods.

21. Equivalent Gate Circuits
2.6 Simplification Theorems
Equivalent Gate Circuits
===
*Read some examples in the text.

22. 2.7 Multiplying Out and Factoring
To obtain a sum-of-products form  Multiplying out using distributive laws
Sum-of-products form:
sum of product terms
consisting of single variables
Still considered to be in sum of product form:
not a single variable
Not in Sum of product form:
*Multiplying out and eliminating redundant terms to get s-o-p forms:

23. 2.7 Multiplying Out and Factoring
To obtain a product of sum form  all sums are the sum of single variable
Product of sum form:
Still considered to be in product of sum form:
*These two make the hardware logic circuits as the two-level
resulting in the same response times.
Sum of Products: OR of AND’s
Product of Sums: AND of OR’s

24. Circuits for SOP and POS form
Sum of product form:
Product of sum form:

25. 2.8 DeMorgan’s Laws DeMorgan’s Laws: to obtain the inverse Proof
X Y
X’ Y’
X + Y
( X + Y )’
X’ Y’ XY
( XY )’
X’ + Y’
0 0
0 1
1 0
1 1
1 1
1 0
0 1
0 0 1
Extension: DeMorgan’s Laws for n variables
Example

26. expression and then complementing each individual variable.
2.8 DeMorgan’s Laws
Inverse of A’B+AB’
A B
A’ B
A B’
F = A’B+AB’
A’ B’
A B
F’ = A’B’ + AB
0 0
0 1
1 0
1 1 1
Dual: ‘dual’ is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0 note: variables are not inverted.
The complement of an expression may be found by making a dual of the entire
expression and then complementing each individual variable.

27. MOS and CMOS
MOSFET (metal-oxide-semiconductor field-effect transistor)
C(Complementary) MOS
General MOSFET symbol
VDD = Power supply voltage,
n channel: positive logic. logic 0 = 0 volt and logic 1 = VDD volt
p channel: negative logic, logic 0 = 0 volt and logic 1 = -Vnn volt. VDS=negative
logic 1 to gate will switch the MOSFET to the ON

28. MOS Inverter
P611, Fig. A.2
For n channel, positive logic case.

29. TTL(Transistor Transistor Logic)
Ref. Digital Electronics, by Uffenbeck
Bipolar Transistor: Base, Emitter, Collector
TR state: cut-off, active, saturation
Vcc
Base
Collector
Emitter

30. Transistor-Transistor-Logic:
Base, Emitter, Collector
off
off

31. *

34. When a high voltage (>2
When a high voltage (>2.7) is applied to input, input current is allowed up to 20 uA.
When a low voltage (<0.4) is applied to input, input current is allowed up to 0.4 mA.

36. Homework
6th Ed.
1, 2, 6, 7,12, 22, 25, 26, 29