ENT 263/4 ELECTRONIC DIGIT
LAB 2: BOOLEAN ALGEBRA AND CIRCUIT
SIMPLIFICATION
OBJECTIVES
1. To implement combinational logic circuits using AND, OR and Inverter gates.
2. To use Boolean theorems and DeMorgan’s theorem in circuit simplification.
EQUIPMENTS/COMPONENTS



Logic gates ( 74XX-series )
Switches
Light Emitting Diodes
INTRODUCTION
Boolean algebra is the mathematical foundation of digital systems. In Boolean algebra
there are three basic operations: OR, AND and NOT which can represent the three basic
logic gates: OR, AND and Inverter gates respectively. In the other words, every Boolean
expression has an equivalent gate description, and vice versa. The combination of logic
gates is called as combinational logic circuit.
In designing a combinational logic circuit, it is highly desirable to find the simplest
implementation – that is, the one with the smallest number of gates or wires. One of the
platforms to simplify the circuit is simplifying the logic expressions by using Boolean
theorems and DeMorgan’s theorem.
DeMorgan's theorem gives a procedure for complementing a complex function. The
complemented expression is formed from the original by replacing all literals by their
complements, and ANDs become ORs and vice versa. This theorem indicates an
interesting relationship between NOR, OR, NAND, and AND:
In this experiment, you will investigate the application of Boolean theorems and
DeMorgan’s theorem in circuit simplification. The combinational logic circuit diagram or
Boolean expression will be given during the laboratory session.
Note: Boolean theorems and DeMorgan theorem are listed in Appendix 2.
KOLEJ UNIVERSITI KEJURUTERAAN UTARA MALAYSIA
ENT 263/4 ELECTRONIC DIGIT
ACTIVITY SHEET
Step 1: Draw a logic diagram for the given expression;
or,
Examine the given logic circuit, and write the Boolean expressions for function
F(A,B,C).
Note: You will be given the combinational logic circuit diagram or the Boolean
expression during the laboratory session.
Q  ( ABC )  ( ABC )  ( ABC )
Step 2: Construct the circuit in Step 1. Connect toggle switches to inputs A, B, C, and a
LED to the circuit output.
Step 3: Construct a truth table for the circuit. Verify the operation of the circuit by setting
the toggle switches to each set of input combinations, and record the output value
observed. Get approval from teaching engineer.
KOLEJ UNIVERSITI KEJURUTERAAN UTARA MALAYSIA
ENT 263/4 ELECTRONIC DIGIT
Step 4: Simplify the circuit using Boolean theorems and DeMorgan’s theorem.
Step 5: Draw the logic diagram for the simplified expression.
KOLEJ UNIVERSITI KEJURUTERAAN UTARA MALAYSIA
ENT 263/4 ELECTRONIC DIGIT
Step 6: Construct the simplified circuit in Step 5. Construct the same truth table in Step 3
and repeat the instruction to record the output value observed for this circuit.
Step 7: Write a simple conclusion (not more than 3 sentences) based on your observation
of the experiment.
KOLEJ UNIVERSITI KEJURUTERAAN UTARA MALAYSIA
ENT 263/4 ELECTRONIC DIGIT
CONCLUSIONS
Answer the following exercises to conclude the experiment.
a) Can all Boolean Sum of Product (SOP) expressions be simplified? If so, state why; if
not, give two examples.
b) Use DeMorgan’s theorem to verify the following statements:
Note: answer should indicate the logic gates diagram and Boolean expressions.
i) NAND gate is equivalent to OR gate with inverted inputs.
ii) NOR gate with inverted inputs is equivalent to AND gate.
iii) NOR gate with inverted inputs is equivalent to AND gate.
iv) NAND gate with inverted inputs is equivalent to OR gate.
KOLEJ UNIVERSITI KEJURUTERAAN UTARA MALAYSIA
ENT 263/4 ELECTRONIC DIGIT
Appendix 2
KOLEJ UNIVERSITI KEJURUTERAAN UTARA MALAYSIA
ENT 263/4 ELECTRONIC DIGIT
Boolean theorems and DeMorgan’s theorem
X
X
X
1
X+0=X
5
X+X=X
2
X+1=1
6
X  X 1
3
X 0=0

7
X X=X
4
X 1=X

8
XX 0
9
XX
10
X+Y=Y+X
11
X Y=Y X
Commutative law
12
X + (Y + Z) = (X + Y) + Z
=X+Y+Z
Associative law
13
X (Y Z) = (X Y) Z
= XYZ
Associative law
14
X(Y + Z) = XY + XZ
Distributive law
15
(W + X)(Y + Z) =
WY + XY + WZ + XZ
Distributive law
16
X + XY = X
17
X  XY  X  Y
18
(X + Y)(X + W) = X + YW

0
X
1

Commutative law

 
1
X
X
 
KOLEJ UNIVERSITI KEJURUTERAAN UTARA MALAYSIA