Lecture 4Introduction to Boolean Algebra
Binary Operators
In the following descriptions, we will let A and B be Boolean variables and define a set of binary operators on them. The term binary in this case does not refer to base-two arithmetic but rather to the fact that the operators act on two operands.
unary operator NOT
binary operators AND, OR, NAND, XOR
Logic Gates
NOT
AND
OR
XOR
NAND
NOR
F(A,B,C) = A + BC'
Truth Tables
A B C C' BC' A+BC'
0 0 0 1 0 0
0 0 1 0 0 0
0 1 0 1 1 1
0 1 1 0 0 0
1 0 0 1 0 1
1 0 1 0 0 1
1 1 0 1 1 1
1 1 1 0 0 1
A Boolean Function Implemented in a Digital Logic Circuit
Pow
er S
uppl
y
Input = 1 1 0
Voltmeter
NOT gates AND gates
OR gate(s)
The Part of the Circuit Usually Not Shown
A One-Bit Adder Circuit
Venn Diagrams
A A B A B
A B A BA B
A
~A + B
A .BA+B
A .BA=B
A A B A B
A B A BA B
A
~A + B
A .BA+B
A .BA .BA=B
Three-Variable Venn Diagram
F(A,B,C) = A + BC'
000
001
010
011
100
101
110
111
A B
C
A B
C
De Morgan's Theorem
BABA
BABA
A B A+B ~(A+B) ~A ~B (~A).(~B) ~(A+B)=(~A).(~B)
0 0 0 1 1 1 1 1
0 1 1 0 1 0 0 1
1 0 1 0 0 1 0 1
1 1 1 0 0 0 0 1
A B A+B ~(A+B) ~A ~B (~A).(~B) ~(A+B)=(~A).(~B)
0 0 0 1 1 1 1 1
0 1 1 0 1 0 0 1
1 0 1 0 0 1 0 1
1 1 1 0 0 0 0 1
Textbook Reading for Chapter 4
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