### ALGEBRA IN COMPUTERS

Algebra is a subject in which the problem
of the sum is solved with the help of letters (I mean the letters of English
language as well as Greek language). It is mostly used by the
mathematicians, engineers, scientist as well as businessmen to solve any
problem of the world. Algebra is very important in many science as
well as commerce studies. So we are going to dig deep in algebra
used in computers.

We will consider Boolean algebra as it is used by the computers to
solve real life problems. So let us begin.

Boolean algebra, like any other deductive
mathematical system , may be with a set of elements , a set of operators and a
number of unproved axioms or postulates. A set of
elements is any collection of objects having a common property.

If S is a set and x is an element of that
set the x€ S denotes that x is an element of that set. A set with
de-numberable (means a small number) number of elements is specified by
braces: A={1, 2, 3, 4} that is the element of set are the number 1 , 2 , 3
,4.

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. As an example consider the relation A * B = C. We
say that * is a binary operator if it specifies a rule for finding c from that
pair (a,b) and also if a,b ,c element of S. However * is not a binary
operator if a, b element of S while the rule finds c not element of S.

The postulates of a mathematical
system form the basic assumptions from which it is possible to deduce the rules
, theorems and property of the system. The most common postulates used to
formulate various algebraic structures are.

1 Closure :- A set S is closed with respect to a
binary operator if, for every pair of elements of S, the binary operator
specifies a rule for obtaining a unique element of S. For example, a set
of natural numbers N = {1,2,3,4......} is closed with respect to the binary
operator plus by the rule of arithmetic addition, since for any a, b element of
N we obtain a unique C element of a + b = c. The set of
natural numbers i not closed with respect to the binary operator minus (-) by
the rules of arithmetic subtraction because 2 - 3 = - 1 and 2, 3 element
of N while (-1) not element of N.

2 Associative law :- A binary operator * on a set S is set to
associative whenever:

(x * y)*z = x * (y * z) for all x, y, z element of S.

3 Commutative law :- A binary operator * on a set S is said to be
commutative whenever :

x * y = y * x for all x ,y element of S

4. Identity element :- A set S is said to have an identity element
with respect to a binary operation * on S if there exist an element e element
of S with the property:

5. Inverse :- A set s having an identity element e with respect to
binary operator * is said to have an inverse whenever, for every x element of S
there exist an element y element s such that:

x * y = e

6. Distributive law :- If * and . are two binary operators on a
set S, * is said to be distributive over . whenever:

x *(y . z) = (x*y).(x*z)

An example of an algebraic structure is a
field. A field is a set of elements, together with two binary operators
each having properties 1 to 5 and both operators give the combination of the
above 5 principles to give the 6 law. The set of real numbers together
with binary operators + and . form the field of real numbers. The field
of real numbers is the basis of for arithmetic and ordinary algebra. The
operators and postulates have the following meanings:

The binary element of + defines addition

The addiditive element is 0.

The additive inverse defines subtraction.

The multiplicative identity is 1.

This is all i got. Thank you for reading.

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