This document discusses Boolean functions and algebraic manipulations of Boolean expressions. It provides examples of simplifying Boolean expressions using Boolean algebra laws and rules. Simplifying expressions can reduce the number of variables and create simpler circuits. One example shows simplifying the expression F = A.B + A.(B+C) + B.(B+C) down to the simplified output of F = B + AC using Boolean algebraic steps, reducing the number of gates from 5 to 2.

1. LECTURE - 8
BOOLEAN FUNCTIONS
&
ALGEBRAIC MANIPULATIONS

2. BOOLEAN FUNCTIONS
■ EXPRESSIONS that contain:
– Binary Variables
■ Values = 0,1
– Binary Operators
■ OR, AND, NOT
– Parenthesis * Optional
– Equal Sign
F = xy’ + x’z

3. F = xy’ + x’y
x y x’ y’ x.y’ x’.y F
0 0 1 1 0 0 0
0 1 1 0 0 1 1
1 0 0 1 1 0 1
1 1 0 0 0 0 0

4. SIMPLIFICATION USING BOOLEAN
ALGEBRA
■ Simplification of Boolean Expressions may be done using:
– Laws / Postulates
– Rules
– Theorems

5. Advantage of Simplification
■ Fewer Variables
■ Simpler Circuit

6. EXAMPLE
F = A.B + A.(B+C) + B.(B+C)
Total No. of Gates = 5
A
B
C
A.B
A.(B+C)
B.(B+C)
F

7. Simplification of
F = A.B + A.(B+C) + B.(B+C)
F = A.B + A.(B+C) + B.(B+C)
= AB + AB + AC + BB + BC
= AB + AC + B + BC
= AB + AC + B (1 + C)
= AB + AC + B (1)
= AB + AC + B
= AB + B + AC
= B (A + 1) + AC
= B (1) + AC
= B + AC

8. SIMPLIFIED OUTPUT:
F = B + AC
Circuit Diagram
A
B
C
A.C
B
F = B + AC
Total No. of Gates = 2

9. MORE EXAMPLES
1. x + x’y
2. x (x’+y)
3. x’y’z + x’yz + xy’
4. xy + x’z + yz
5. (x+y).(x’+z).(y+z)
6. y (wz’ +wz) + xy

10. SOLUTIONS TO EXAMPLES
1. x+y
2. xy
3. x’z+xy’
4. xy+x’z
5. x’y+xz
6. y (x+w)

11. CIRCUIT DIAGRAMS
HOMEWORK:
Draw the Circuit Diagrams of the
examples