Page 56 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Unit 3: Boolean Algebra

Notes
Structure:
• Basic Properties • Derived Properties
• Boolean Functions • Canonical Form
• Boolean Addition • Boolean Multiplication
• Electrical Switching System • Circuits with Composite Operations

ABC+ ABC+ ABC+ ABC
st th
Factoring BC out of 1 and 4 terms
BC(A+A)+ ABC+ ABC
Applying identity A+A= 1
BC(1)+ ABC+ ABC

Applying identity 1A =A
BC+ ABC+ ABC
st rd
FactoringB out of 1 and 3 terms
B(C+ AC)+ ABC
Applying rule A+ AB =A+ B to the
C+ AC term
B(C+A)+ ABC
Distributing terms
BC+AB+ ABC
nd rd
Factoring A out of 2 and 3 terms
BC+ A(B+ BC)
Applying rule A+ AB =A+ B to the
B+BC term
BC+ A(B+C)
Distributing terms
BC+AB+AC
or Simplified result
AB+BC+AC
There are three basic operations in Boolean algebra AND, OR and NOT. These are symbolized
by ∩, U and {} respectively in case of the theory of sets. In this unit the more common symbolic
plus+, dot ‘.’ and prime ( ′ ) would be used for the three operations respectively. The similarity
would become obvious in the way the present chapter would synthesize and generalize what
we have studied earlier and apply it to the end purpose of the designing of the electric circuits.
Questions:
1. Who was the founder of Boolean algebra?
2. Explain the basic structures of Boolean algebra.

Self Assessment

Choose the correct answer:
+
5. One of DeMorgan’s theorems states that x + y = xy and xy . = xy . Simply stated, this
means that logically there is no difference between:
(a) a NOR and an AND gate with inverted inputs

LOVELY PROFESSIONAL UNIVERSITY 51