Boolean Algebra

Boolean Algebra:
• Boolean algebra is the basic mathematics needed
for the study of logic design of digital systems.
• In 1854, George Boole an English mathematician
gave the concept of “Logical algebra” known
today as Boolean algebra.
• Boolean algebra is a convenient and systematic
way of expressing and analyzing the operation of
logic circuits.
• Claude Shannon was the first to apply Boole’s
work to the analysis of relays and switching
circuits in 1938.was Other Logic Operations
• In 1904, Huntington gave a set of postulates that
form the basis of formal definition of Boolean
algebra.
Set Notations:

Mathematical systems can be defined with:
– A set of elements; A set of elements is any collection of
objects having a common property.
– A set of operators; A binary operator defined on a set S of
elements is a rule that assigns to each pair of elements from
S a unique element from S.
– A number of unproved axioms or postulates that form the
basic assumptions from which it is possible to deduce the
rules, theorems and properties of the system.
• The following notations are being used in this class:
–      x∈S indicates that x is an element of the set S.
–      y ∉ S indicates that y is not an element of the set S.


–    A = {1, 2, 3, 4} indicates that set A exists with a finite number
of elements (1, 2, 3, 4).
Basic Postulates:

The basic postulates of a mathematical system are:
– Closure. A set S is closed w.r.t a binary operator if this operation
only produces results that are within the set of elements defined by
the system.
– Associative Law. A binary operator is said to be associative when:
» (x * y) * z = x * (y * z)
– Commutative Law. A binary operator is said to be commutative
when:
» x * y = y * x
– Identity Element. A set is said to have an identity element with
respect to a binary operation if there exists an element, e, that is a
member of the set with the property:
» e * x = x * e = x for every element of the set
• Additive identity is 0 and multiplicative identity is 1
– Note: * + and . are binary operators