The document provides an overview of Boolean algebra, describing it as a mathematical technique for simplifying logic expressions to create more efficient logic circuits. It outlines key concepts such as Boolean constants, variables, functions, and theorems, including laws of commutativity, associativity, and distributivity. Additionally, it presents several Boolean theorems and their applications in logic circuit design.

1. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
Affiliated Institution of G.G.S.IP.U, Delhi
DIGITAL ELECTRONICS
Paper Code :- BCA 106
Keywords:
Boolean Algebra, logic, Laws
By :-HARI MOHAN JAIN

2. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
2
Boolean
Algebra

3. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
3
What is Boolean Algebra ?
Boolean Algebra is a mathematical technique that provides the ability to
algebraically simplify logic expressions.
These simplified expressions will result in a logic circuit that is equivalent to
the original circuit, yet requires fewer gates.

4. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
4
Boolean Algebra
• Boolean Constants
– these are ‘0’ (false) and ‘1’ (true)
• Boolean Variables
– variables that can only take the vales ‘0’ or ‘1’
• Boolean Functions
– each of the logic functions (such as AND, OR and NOT)
are represented by symbols as described above
• Boolean Theorems
– a set of identities and laws – see text for details

5. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
5
00X 
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
X1X  XXX  0XX 
Boolean Theorems (1 of 7)
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
Single Variable - AND Function
Theorem #1 Theorem #2 Theorem #3 Theorem #4

6. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
6
X0X  11X 
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
XXX  1XX 
Boolean Theorems (2 of 7)
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
Single Variable - OR Function
Theorem #5 Theorem #6 Theorem #7 Theorem #8

7. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
7
XX 
0 1 0
1 0 1
Single Variable – Invert (NOT) Function
Boolean Theorems (3 of 7)
X XX
Theorem #9

8. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
8
Boolean Theorems (4 of 7)
XYYX 
Commutative Law
Theorem #10A – AND Function

9. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
9
Boolean Theorems (4 of 7)
XYYX 
Commutative Law
Theorem #10B – OR Function

10. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
10
Boolean Theorems (5 of 7)
ZY)(XZ)(YX 
Associative Law
Theorem #11A – AND Function

11. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
11
Boolean Theorems (5 of 7)
ZY)(XZ)(YX 
Associative Law
Theorem #11B – OR Function

12. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
12
Boolean Theorems (6 of 7)
YZYWXZXWZ)Y)(WX( 
Distributive Law
Theorem #12B – OR Function

13. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
13
Boolean Theorems (7 of 7)
YXYXX 
YXYXX 
YXYXX 
YXYXX 
Consensus Theorem
Theorem #13A
Theorem #13B
Theorem #13C
Theorem #13D