ALGEBRA IN COMPUTERS

 

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 EVERGREEN 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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