Experiment 4
BOOLEAN ALGEBRA THEOREMS
OBJECTIVE
The student will be able to do the following:
a.
b.
c.
d.
Identify the different Boolean Algebra Theorems and its properties.
Plot circuits and prove De Morgan’s Theorem equivalence.
Construct circuits and prove Distributive Law equivalence.
Simplify circuit equation by manipulation using boolean equations.
Boolean Algebra
Boolean algebra is used for two-valued logic that is present on any digital
system. Named after in the honor of English Mathematician George Boole,
Boolean algebra describes the interconnection of digital gates and how
simplification can be implemented through its use.
Table 1 present the properties of Boolean algebra theorems. The first
three theorems state the properties of Boolean operations AND, OR, and
NOT. Theorem 3a states ORing logic-1 with anything will always result a
logic-1.
Idempotent law (fourth theorem) states that repetitions of variables in
an expression are redundant and may be deleted.
Involution law produces a cancellation effect
complementation occurs as stated on Theorem 6.
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when double
Interchanging the order of variables does not change the result of the
operation as stated in Commutative law. Theorems 8 and 9 show
simplification of Boolean expression. De Morgan’s law the effect of
complementation on variables when connected by the AND and OR
operations.
Any order in groupings can be applied using Associative law when
ANDing and ORing of variables. Distributive law shows how factoring is
done using the same principle in algebra.
Take note of the symmetrical property of Boolean algebra equations.
This is known as the principle of duality. AND and OR operation (and vice
versa) can be interchanged on each occurrence.
Equation Complementation
The complement of an equation is obtained by the interchange of 1’s to 0’s
and 0’s to 1’s. To achieve this, we can apply algebraically by using De
Morgan’s theorem. The generalized form of this law states that the
complement of an expression is obtained by interchanging AND and OR
operations and complementing each variable each variable and constant.
Let us apply complementation on the following:
EQUIPMENT
Prototyping board (breadboard)
DC Power Supply 5V or 9V Battery
Light Emitting Diode (LED) (4)
Solid-core wire (gauge 22, 1 meter long)
Digital ICs:
7404
Hex Inverter
7408 (2)
Quad AND
7432 (2)
Quad OR
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PROCEDURE
De Morgan’s Law
1. Construct Circuit 1 on your prototyping board. Take note of the number
assigned inside the logic gate symbols. This denotes the IC number package
designation for each IC that you will use.
2. Write the equivalent logic equation of Circuit 1. ______________________
3. Construct circuit 2.
4. Write the equivalent logic equation of Circuit 2. _____________________
5. Test the different input combinations of Circuit 1 and Circuit 2 and fill-up the
following truth tables.
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6. Do the two circuits equal in terms of output D? ________________________
7. If you were to choose between Circuit 1 and Circuit 2, which design will you
implement and why? _______________________________________________
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8.
Simplify Circuit 1 equation using De Morgan’s theorem. Show your step-bystep solution.
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Distributive Law
9. Construct Circuit 3.
10. Write the equivalent equation of Circuit 3. ___________________________
11. Construct Circuit 4.
12. Write the equivalent equation of Circuit 4. ___________________________
13. Test the input combinations of Circuit 3 and 4 and fill up the following truth
tables.
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14. If you were to choose between Circuit 3 and Circuit 4, which design will you
implement and why? _______________________________________________
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15. Simplify Circuit 3 equation using Distributive law. Show your step-by-step
solution. Hint: Apply the theorem on the shaded portion of Circuit 3.
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REFERENCES
Mano, M. and Kime, C. (2001). Logic and Computer Design Fundamentals (2nd
Edition Updated). Prentice-Hall International
Givone, D. and Roesser, R. (1980).
Introduction. McRaw-Hill International.
Concept by:
JERRY C. ESPERANZA
Copyright 2009
http://ThrivingAndLiving.blogspot.com
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Microprocessors/Microcomputers An