Boolean Algebra and
Digital Logic
Objectives
• Understand the relationship between Boolean
logic and digital computer circuits.
• Learn how to design simple logic circuits.
• Understand how digital circuits work together to
form complex computer systems.
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Introduction
• In the latter part of the nineteenth century, George
Boole incensed philosophers and mathematicians
alike when he suggested that logical thought could
be represented through mathematical equations .
– How dare anyone suggest that human thought could be
encapsulated and manipulated like an algebraic formula?
• Computers, as we know them today, are
implementations of Boole’s Laws of Thought.
– John Atanasoff and Claude Shannon were among the
first to see this connection.
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Introduction
• In the middle of the twentieth century, computers
were commonly known as “thinking machines” and
“electronic brains.”
– Many people were fearful of them.
• Nowadays, we rarely ponder the relationship
between electronic digital computers and human
logic. Computers are accepted as part of our lives.
– Many people, however, are still fearful of them.
• In this chapter, you will learn the simplicity that
constitutes the essence of the machine.
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Boolean Algebra
• Boolean algebra is a mathematical system
for the manipulation of variables that can
have one of two values.
– In formal logic, these values are “true” and “false.”
– In digital systems, these values are “on” and “off,”
1 and 0, or “high” and “low.”
• Boolean expressions are created by
performing operations on Boolean variables.
– Common Boolean operators include AND, OR, and
NOT.
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Boolean Algebra
• A Boolean operator can be
completely described using a
truth table.
• The truth table for the Boolean
operators AND and OR are
shown at the right.
• The AND operator is also known
as a Boolean product. The OR
operator is the Boolean sum.
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Boolean Algebra
• The truth table for the
Boolean NOT operator is
shown at the right.
• The NOT operation is
most often designated by
an overbar. It is
sometimes indicated by a
prime mark ( ‘ ) or an
“elbow” ( ¬).
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Boolean Algebra
• A Boolean function has:
• At least one Boolean variable,
• At least one Boolean operator, and
• At least one input from the set {0,1}.
• It produces an output that is also a member of
the set {0,1}.
Now you know why the binary numbering
system is so handy in digital systems.
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Boolean Algebra
• The truth table for the
Boolean function:
is shown at the right.
• To make evaluation of the
Boolean function easier,
the truth table contains
extra (shaded) columns
to hold evaluations of
subparts of the function.
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Boolean Algebra
• As with common
arithmetic, Boolean
operations have rules of
precedence.
• The NOT operator has
highest priority, followed
by AND and then OR.
• This is how we chose
the (shaded) function
subparts in our table.
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Boolean Algebra
• Digital computers contain circuits that implement
Boolean functions.
• The simpler that we can make a Boolean function,
the smaller the circuit that will result.
– Simpler circuits are cheaper to build, consume less
power, and run faster than complex circuits.
• With this in mind, we always want to reduce our
Boolean functions to their simplest form.
• There are a number of Boolean identities that
help us to do this.
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3.2 Boolean Algebra
• Most Boolean identities have an AND (product)
form as well as an OR (sum) form. We give our
identities using both forms. Our first group is rather
intuitive:
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Boolean Algebra
• Our second group of Boolean identities should be
familiar to you from your study of algebra:
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Boolean Algebra
• Our last group of Boolean identities are perhaps
the most useful.
• If you have studied set theory or formal logic, these
laws are also familiar to you.
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Boolean Algebra
• Sometimes it is more economical to build a
circuit using the complement of a function (and
complementing its result) than it is to implement
the function directly.
• DeMorgan’s law provides an easy way of finding
the complement of a Boolean function.
• Recall DeMorgan’s law states:
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Boolean Algebra
• DeMorgan’s law can be extended to any number
of variables.
• Replace each variable by its complement and
change all ANDs to ORs and all ORs to ANDs.
• Thus, we find the the complement of:
is:
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Boolean Algebra
• Through our exercises in simplifying Boolean
expressions, we see that there are numerous
ways of stating the same Boolean expression.
– These “synonymous” forms arelogically equivalent.
– Logically equivalent expressions have identical truth
tables.
• In order to eliminate as much confusion as
possible, designers express Boolean functions in
standardized or canonical form.
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Boolean Algebra
• There are two canonical forms for Boolean
expressions: sum-of-products and product-of-sums.
– Recall the Boolean product is the AND operation and the
Boolean sum is the OR operation.
• In the sum-of-products form, ANDed variables are
ORed together.
– For example:
• In the product-of-sums form, ORed variables are
ANDed together:
– For example:
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Boolean Algebra
• It is easy to convert a function
to sum-of-products form using
its truth table.
• We are interested in the
values of the variables that
make the function true (=1).
• Using the truth table, we list
the values of the variables that
result in a true function value.
• Each group of variables is
then ORed together.
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Logic Gates
• We have looked at Boolean functions in abstract
terms.
• In this section, we see that Boolean functions are
implemented in digital computer circuits called gates.
• A gate is an electronic device that produces a result
based on two or more input values.
– In reality, gates consist of one to six transistors, but
digital designers think of them as a single unit .
– Integrated circuits contain collections of gates suited to a
particular purpose.
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Logic Gates
• The three simplest gates are the AND, OR, and NOT
gates.
• They correspond directly to their respective Boolean
operations, as you can see by their truth tables.
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Logic Gates
• Another very useful gate is the exclusive OR
(XOR) gate.
• The output of the XOR operation is true only when
the values of the inputs differ.
Note the special symbol ⊕
for the XOR operation.
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Logic Gates
• NAND and NOR
are two very
important gates.
Their symbols and
truth tables are
shown at the right.
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Logic Gates
• NAND and NOR
are known as
universal gates
because they are
inexpensive to
manufacture and
any Boolean
function can be
constructed using
only NAND or only
NOR gates.
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Logic Gates
• Gates can have multiple inputs and more than
one output.
– A second output can be provided for the complement
of the operation.
– We’ll see more of this later.
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