A Boolean Algebra
352A.1 INTRODUCTION
Like any other mathematical system, Boolean algebra, is defined by a set of elements X, a set
of operators Y, and a set of postulates (axioms) that defines the properties of X and Y. A set is
a collection of objects sharing a common property. Set of operators Y = {AND, OR, NOT},
where AND and OR are binary operators represented by ‘. ’ and ‘+’ respectively, and NOT is
a unary operator represented by ‘ ′ ’. The postulates are the basic assumptions of the algebraic
structures on which it is possible to construe the rules and theorems of the system. The postulates
need no proof. It provides the basis to the theorems. Here we summarize some of the common
postulates used by various algebraic structures:
I.Closure. The operators ‘. ’ and ‘+’ are closed, which means that if x and y ∈ X,
then x + y and x. y are also ∈ X and unique.
II.Commutative. Binary operators ‘. ’ and ‘+’ are commutative on a set X such that
if x and y ∈ X, then we have,
x + y = y + x and x. y = y. x
III.Associative. Binary operators ‘. ’ and ‘+’ are associative on a set X such that if x, y
and z ∈ X, then we have,
x + (y + z) = (x + y) + z and x. (y. z) = (x. y). z
IV. Existence of an Identity Element. There exists an identity element ê for ∀x ∈ X
with respect to binary operators ‘. ’ and ‘+’ i.e.
x + ê = ê + x = x and x. ê = ê. x = x
For example, 0 is the identity element for algebraic structure (I, +), where I is the set of
integers and ‘+’ is the binary addition operation of integers i.e., x + 0 = 0 + x = x, for ∀x ∈ I.
Therefore, element 0 is called additive identity. (Reader self verify that element 1 will be a
multiplicative identity).
V.Existence of an Inverse Element. There exists an inverse element y ∈ X for ∀x ∈ X
with respect to binary operators ‘. ’ and ‘+’ i.e.,
x + y = ê and x. y = ê
For example, the additive inverse define the subtraction s.t. x + (– x) = ê. Similarly, the
multiplication inverse defines division s.t. x. (1/x) = ê.
VI.Distributive Property. Since ‘. ’ and ‘+’ are two binary operators on set X, and if x,
y and z ∈ X then operator ‘. ’ is said to be distributive over operator ‘+’ whenever,
x. (y + z) = x. y + x. z
Appendix