### DIGGING DEEP IN ALGEBRA USED IN COMPUTERS

**DIGGING DEEP IN ALGEBRA USED IN COMPUTERS**

Now I will show you how our algebra
is used in computers. I will take
Boolean algebra as example because it is widely used in computers. So let’s begin.

In 1854 George Bole created Boolean
algebra, which was a systematic treatment of logic for algebraic purpose or
making them look like a language. Then
in 1938 C.E. Shannon introduced a two valued Boolean algebra and called it
switching algebra, in which he demonstrated that the properties of bitable
electrical switching circuits can be represented by this algebra. For the formal definition of Boolean algebra,
we shall employ the postulates formulated by E.V. Huntington.

Now continuing with that lets
switch with the topic. Other sets
of postulates have been used. Boolean algebra is an algebraic structure defined
on a set of elements B together with two binary operators + and * provided the
following (Huntington postulates
are satisfied . But the problem is that
some misconception needed to be cleared out, it is not same as ordinary
algebra. The reason is given below

·
Huntington
postulates do not include the associative law.
However, this law holds for Boolean algebra and can be derived (for both
operators) from the other postulates.

·
The distributive law of + and over * that is x +
(y*z) =
(x + y) * (y + z) is valid for Boolean algebra but not for ordinary
algebra.

·
Boolean algebra does not have additive or
multiplicative inverses; therefore there are no subtraction or division
operations.

·
Ordinary algebra deals with the real numbers,
which constitute an infinite set of elements.
Boolean algebra deals with the as yet undefined set of elements B, but
in the two valued algebra defined such as 0 and 1.

Boolean algebra resembles ordinary
algebra in some respects. The choice of
symbols + and * is intentional to facilitate Boolean algebraic manipulations by
person already familiar with ordinary algebra.
Although one can use some knowledge from ordinary algebra to deal with
Boolean algebra, the beginner must be careful not to substitute the rules of
ordinary algebra where they are not applicable.

It is important to distinguish
between the elements of the set of an algebraic structure and the variables of
an algebraic system. For example, the
elements of the field of real numbers are numbers where as variables such as a,
b, c, etc, are used in ordinary algebra, are symbols that stand for real
numbers. Similarly in Boolean algebra,
one defines the elements of the set B,
and variables such as x,y, z are merely a symbols that represent the
elements. At this point, it is important
to realize that in order to have a Boolean algebra, one must show:

·
The elements of the set B,

·
The rules of operation for the two binary
operators and

·
That the set of elements B, together with the
two operators satisfies the six Huntington
postulates.

One can formulate many Boolean algebra,
depending on the choice of elements of B and the sets of operations.

Thanks for reading

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