Boolean Algebra and Reduction Techniques Lecture 4 Digital Electronics

1.

Boolean Algebra and Reduction
Techniques
Lecture 4
Digital Electronics

2.

Course Materials
All needed Software and course materials will
be located on Canvas.
◻ Materials that are used in this slides are taken
from the textbook “Digital Electronics A
Practical Approach with VHDL” by William
Kleitz
◻

3.

Boolean Algebra Laws and
Rules

4.

Review Questions
The output of an AND gate with one of its
inputs connected to 1will always output a
level equal to the level at the other input.
True or false?
How many gates are required to implement
the following Boolean equations?
1.
2.
a)
b)
c)
X = (A + B)C
Y = AC + BC
Z = (ABC + CD)E

5.

Simplification Using Boolean
Algebra
equivalent circuit can be formed with fewer
gates or fewer inputs
◻ cost of the circuit is reduced and its reliability is
improved
◻

6.

7.

Simplification Using Boolean
Algebra
Example. Simplify the following logic circuit.

8.

Simplification Using Boolean
Algebra

9.

10.

11.

De Morgan’s Theorem
To simplify circuits
containing NANDs
and NORs
◻ convert an
expression having
an inversion bar
over two or more
variables into an
expression having
inversion bars over
single variables
◻
Simplify

12.

De Morgan’s Theorem
Example. Write the Boolean equation for the
circuit. Use De Morgan’s theorem and then
Boolean algebra rules to simplify the equation.
Draw the simplified circuit.

13.

De Morgan’s Theorem
Solution.

14.

The Universal Capability of
NAND and NOR Gates
◻
all the other logic gates (inverter, AND, OR,
NOR) can be formed

15.

Karnaugh Mapping
1.
Transform the Boolean equation to be reduced into an SOP
expression
2.
Fill in the appropriate cells of the K-map
3.
Encircle adjacent cells in groups of two, four, or eight. (The more
adjacent cells encircled, the simpler the final equation is; adjacent
means a side is touching, not diagonal.)
4.
Find each term of the final SOP equation by determining which
variables remain constant within each circle

16.

17.

18.

19.

Review Questions
1.
2.
3.
4.
Using De Morgan’s theorem, you can prove
that a NOR gate is equivalent to an
___________ (OR, AND) gate with inverted
inputs.
Why are NAND gates and NOR gates
sometimes referred to as universal gates?
The number of cells in a Karnaugh map is
equal to the number of entries in a
corresponding truth table. True or false?
Adjacent cells in a Karnaugh map are
encircled in groups of 2, 4, 6, or 8. True or

20.

21.

Simplify the following equations using the Karnaugh
mapping procedure:

22.

Q&A
Any Questions?