### Lecture 8 (K-Map)

COMPUTER FUNDAMENTALS
David Samuel Bhatti
[email protected]
Boolean Operations and Expressions
Boolean Operations and Expressions

0+0=0
0+1=1
1+0=1
1+1=1

Multiplication
0*0=0
0*1=0
1*0=0
1*1 =1
Laws and Rules of Boolean Algebra
Laws Boolean Algebra



Commutative Laws
Associative Laws
Distributive Law
Laws of Boolean Algebra

A+B=B+A
Laws of Boolean Algebra

Commutative Law of Multiplication:
A*B=B*A
Laws of Boolean Algebra

A + (B + C) = (A + B) + C
Laws of Boolean Algebra

Associative Law of Multiplication:
A * (B * C) = (A * B) * C
Laws of Boolean Algebra

Distributive Law:
A(B + C) = AB + AC
Rules of Boolean Algebra
Rules of Boolean Algebra

Rule 1
OR Truth Table
Rules of Boolean Algebra

Rule 2
OR Truth Table
Rules of Boolean Algebra

Rule 3
AND Truth Table
Rules of Boolean Algebra

Rule 4
AND Truth Table
Rules of Boolean Algebra

Rule 5
OR Truth Table
Rules of Boolean Algebra

Rule 6
OR Truth Table
Rules of Boolean Algebra

Rule 7
AND Truth Table
Rules of Boolean Algebra

Rule 8
AND Truth Table
Rules of Boolean Algebra

Rule 9
Rules of Boolean Algebra

Rule 10: A + AB = A
AND Truth Table
OR Truth Table
Rules of Boolean Algebra

Rule 11:
A  AB  A  B
AND Truth Table
OR Truth Table
Rules of Boolean Algebra

Rule 12: (A + B)(A + C) = A + BC
AND Truth Table
OR Truth Table
DeMorgan’s Theorem
DeMorgan’s Theorems

Theorem 1
XY  X  Y

Theorem 2
X  Y  XY
Remember:
“Break the bar,
change the sign”
Standard Forms of Boolean Expressions
Standard Forms of Boolean Expressions

The sum-of-product (SOP) form
Example: X = AB + CD + EF

The product of sum (POS) form
Example: X = (A + B)(C + D)(E + F)
The Karnaugh Map
The Karnaugh Map
4-Variable Karnaugh Map
The Karnaugh Map
4-Variable Example
Special thanks to Dr P. K. Sinah for using slides
Simplification By Karnaugh Map
3-Variable Karnaugh Map
Simplification By Karnaugh Map
3-Variable Karnaugh Map
Simplification By Rules
Quiz
1. Compute (255-256)10 using Compliment in
binary mode
2. True/False
1.
2.
3.
4.
5.
Base 2 Number System={0,1}……..T/F
Base 4 Number System={1,2,3,4}……..T/F
Base 8 Number System={0,1,2,3,4,5,6,7,8}……..T/F
Base 16 Number
System={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}……..T/F
F=16 in Base 16 Number System
Special thanks to Dr P. K. Sinah for using slides
Key Terms

AND
OR
NOT
XOR
NAND
SOP
POS

Logic Gate

K-Map






Special thanks to Dr P. K. Sinah for using slides