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Unit 3: Boolean Algebra

In written form, the complement of “A” denoted as “A-not” or “A-bar”. Sometimes a “prime” Notes
symbol is used to represent complementation (‘A’). Boolean complementation finds equivalency
in the form of the NOT gate.
The basic definition of Boolean quantities has led to the simple rules of addition and multiplication,
and has excluded both subtraction and division as valid arithmetic operations. We have symbols
for denoting Boolean variables, and their complements.

Remember that in the world of Boolean algebra, there are only two possible
values for any quantity and for any arithmetic operation: 1 or 0.

3.1.2 Boolean Algebraic Identities
In mathematics, an identity is a statement true for all possible values of its variable or variables.
The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original
“anything,” no matter what value that “anything” (x) may be. Like ordinary algebra, Boolean
algebra has its own unique identities based on the bivalent states of Boolean variables. If A is a
Boolean variable, Figure 3.2 below shows the basic Boolean algebraic identities.
Figure 3.2: Basic Boolean Algebraic Identities

Additive Multiplicative
A + 0 + = A 0A = 0
A + 1 = 1 1A = A
A + A = A AA = A

A + A = 1 AA = 0

3.1.3 Boolean Algebraic Properties
Another type of mathematical identity, called a “property” or a “law,” describes how differing
variables relate to each other in a system of numbers. Assuming A and B are Boolean numbers,
Figure 3.3 lists the Boolean algebraic properties.

Figure 3.3: Basic Boolean Algebraic Properties of Additive Association,
Multiplicative Association and Distribution
Additive Multiplicative

A + B = B + A AB = BA
A + (B + C) = (A + B) + C A(BC) = (AB) C
Distributive A(B + C) = AB + AC

3.1.4 Translating Truth Tables into Boolean Expressions
In designing digital circuits, the designer often begins with a truth table describing what the
circuit should do. The design task is largely to determine what type of circuit will perform the
function described in the truth table. While some people seem to have a natural ability to look
at a truth table and immediately envision the necessary logic gate or relay logic circuitry for the
task, there are procedural techniques available for the rest of us. Here, Boolean algebra proves
its utility in a most dramatic way.

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