DIGGING DEEP IN ALGEBRA USED IN COMPUTERS


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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.


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