Electronic Systems &
Digital Electronics
Lecture 2
Assoc. Prof. Roaa Mubarak
The course includes:
1. Introduction to digital systems.
2. Numbering system and Codes.
3. Boolean Switching Algebra & logic gates.
4. Combinational Logic Circuits.
5. Combinational Circuit Implementation.
6. Asynchronous Sequential Circuits.
7. Synchronous Sequential Circuits.
8. Memory.
Boolean Switching Algebra & logic gates
Logic Gates
Logic Gates
Boolean Algebra
• What is the difference between Boolean algebra and arithmetic
algebra?
The first difference is that in Boolean algebra we have only the (+)
and (*) operators, but we do not have subtraction (-) or division (/)
like in mathematics.
There are three fundamental logical operations, from which all
other functions, no matter how complex, can be derived.
These Basic functions are named: AND - OR - NOT (INVERTER).
Boolean Algebra
Boolean Algebra
• Example1:
• Example2:
Boolean Algebra
• Example 3:
Boolean Algebra
• Example 4:
Boolean Algebra
• Example 5:
Show that X’Y’Z + YZ +XZ = Z
X’Y’Z + YZ +XZ = (X’Y’ + Y + X) Z
= ((X’+Y)(Y’+Y) +X) Z
= ((X’+Y) + X) Z
= (X’+ X +Y) Z
= (1 +Y) Z
= Z
Boolean Algebra
• Example 6:
Prove that XY +X’Z +YZ = XY +X’Z
XY +X’Z +YZ = XY +X’Z +YZ (X + X’)
= XY +X’Z +YZX + YZX’
= XY +X’Z +XYZ + X’ZY
= XY (1 +Z) +X’Z (1+Y)
= XY +X’Z
Boolean Algebra
• Example 7:
Find the Complement of F= (X’+Y)(XZ+YZ’)
F’= [(X’+Y)(XZ+YZ’)]’
= (X’+Y)’+(XZ+YZ’)’
= X’’. Y’ + (XZ)’(YZ’)’
= XY’ + (X’+Z’)(Y’+Z)
How to describe A logic System?
By using one of the following two methods:
• A Truth Table
• A Boolean Expression
Truth Table
• A Truth Table is a table of combinations of the binary variables showing the
relationship between the different values that the input variables take and
the result of the operation (output).
• The number of rows in the Truth Table is 2n , where n = number of input
variables in the function. The binary combinations are obtained from the
binary number by counting from 0 to 2n – 1.
Truth Table
• Canonical Form – In Boolean algebra, Boolean function can be expressed as
Canonical Disjunctive Normal Form known as minterm and some are
expressed as Canonical Conjunctive Normal Form known as maxterm.
Truth Table
• In Minterm, we look for the functions where the output results in “1” while in
Maxterm we look for function where the output results in “0”.
We perform Sum of minterm also known as Sum of products (SOP) .
We perform Product of Maxterm also known as Product of sum (POS).
Boolean functions expressed as a sum of minterms or product of maxterms are
said to be in canonical form.
Truth Table
• A Boolean function can be expressed algebraically from a given truth table by
forming a :
• minterm for each combination of the variables that produces a 1 in the
function and then taking the OR of all those terms.
• Maxterm for each combination of the variables that produces a 0 in the
function and then taking the AND of all those terms.
Sum of Product
Product of Sum
How to get the Boolean expression
Getting Boolean Expression from:
1. Truth table
How to get the Boolean expression
Getting Boolean Expression from:
2. Logic Circuit
How to get the Boolean expression
Getting Boolean Expression from:
3. Karnough Map
In many digital circuits and practical problems, we need to find expressions
with minimum variables. We can minimize Boolean expressions of 3, 4, 5 and 6
variables very easily using K-map without using any Boolean algebra theorems.
K-map can take two forms:
• Sum of product (SOP)
• Product of Sum (POS)
