devotion & mathematics

  Problem :        If G is the set of even integers i.e. G = { ...,-4,-2,0,2,4,...} then prove that G is an abelian group      with  usual addition .                                               Or      Show that the set of all even integers forms an abelian group under ordinary '+' . Solution :            Suppose G is the set of even integers i.e. G = { ...,-4,-2,0,2,4,...}.           Now we prove (G,+) is abelian. Closure Property :             Let a, b 𝞊 G.            Suppose a = 2x and b = 2y for x , y 𝞊 Z.             Now a + b = 2x+2y                              =  2( x+y) | sinc...

Basic Definitions :  * Binary Operation  * Algebraic Structure * Quasi- Group or Groupoid * Semi Group * Monoid * Group * Abelian Group Binary Operation :            Let S be a non-empty set . If f : SxS→R is a mapping , then f is called binary                   operation or binary  composition on S.           Thus        If a relation in S is such that every pair ( distinct or equal ) of elements of S  taken in definite  order is associated with a unique element of S then it is called a binary operation in S. Otherwise the relationis not binary operation in S and the relation is simply an operation in S.      (a,b) 𝟄 SxS , ∃ a unique element f(a,b) 𝟄 S. We observe that addition, multiplication, subtraction are binary operations in R and division is not a binary operation in R  why because division by 0 is not defined...