Introduction to Digital Electronics – Problems 4 – Boolean Algebra
Problem Sets for Module 4: Boolean Algebra and
Simplification
PROBLEM 4.1: The NOT Operator
In a truth table, Y = all of the places that the output goes to a logical
1. Not Y or Y with a line over the top equals all of the places a truth
table goes to a logical 0.
Given the SOP expression below for Y, Write the SOP expression for
NOT Y.
_
_ _
_ _
Y= ABC + ABC + ABC + ABC
Fill in the truth table, and write down the Boolean algebra SOP
expression for NOT Y.
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
|
|
|
|
|
|
|
|
|
Y
_
Y =
___________________
PROBLEM 4.2: The Commutative and the Associative Laws
For the Boolean algebra expressions below, tell which ones are legal
operations and which ones are not:
A.
A+B+C = B+A+C
C.
AB(DC) =DCBA
E.
B.
ABCD = B(A+C)
D.
ABD+BCE+BDE =B(AD+CE+DE)
BC(ADE ) = BD(ACE)
1
Introduction to Digital Electronics – Problems 4 – Boolean Algebra
2
PROBLEM 4.3: Basic Boolean Algebra Rules
The basic Boolean Algebra rules are sometimes expressed in simple
rules of the laboratory. For each laboratory design rule expressed
below, identify which basic Boolean algebra rule applies.
A.
Tie unused AND and NAND gate inputs high. Rule____
B.
Ground unused OR or NOR gate inputs. Rule_____
C.
Inverting a logic signal twice has not effect on the output
Rule______
level.
D.
An AND gate with the same variable on the inputs, yields
variable at the output. Rule_________
that
E.
An OR gate with the same variable on the inputs, yields
variable at the output. Rule___________
that
F.
An OR gate with a variable and the inverse of the same
variable on its inputs yields a logical 1 at the output.
Rule__________
PROBLEM 4.4: DeMorgan’s Theorem
Use DeMorgan’s Theorem to simplify each of the following expression
and to convert them to SOP form:
PROBLEM 4.5: Boolean Algebra Simplification
Use the rules and theorems of Boolean algebra to simplify each of the
following expressions:
Introduction to Digital Electronics – Problems 4 – Boolean Algebra
3
PROBLEM 4.6:
Combined Boolean Algebra and DeMorgan’s
Theorem Simplification.
In the following problems, use a combination of Boolean algebra rules
and DeMorgan’s Theorem to obtain the simplest form of the equation:
PROBLEM 4.7:
Designing With Truth Tables and Boolean
algebra
For each of the truth tables below, write the unsimplified Boolean
algebra expressions:
A.
ABCD
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
X
1
0
0
1
0
0
1
0
0
1
0
0
0
1
0
0
B.
RST
000
001
010
011
100
101
110
111
X
0
0
1
0
1
1
0
1
Introduction to Digital Electronics – Problems 4 – Boolean Algebra
4
PROBLEM 4.8:
Designing With Truth Tables and Boolean
algebra
For each of the truth tables above, reduce the expressions to their
simplest forms using Boolean algebra.
PROBLEM 4.9:
Reducing Truth Tables using Karnaugh Maps
For the truth tables shown in problem 4.7, use a K map to reduce the
Boolean algebra equations to their simplest forms. Compare the result
to the results of problem 4.8.
PROBLEM 4.10: Karnaugh map Simplification
Simplify each of the following K maps and write the reduced Boolean
algebra equations: