devotion & mathematics

  Problem :           Show that the set Q + of all +ve rational numbers forms an abelian group under the composition           defined by aob = (ab) / 3 for a,b 𝞊  Q + Solution :          Suppose Q +   is the set of all +ve rational numbers.               Define the operation ' o ' on  Q +   by  aob = (ab) / 3 for a,b 𝞊  Q + .           Now we are going to prove (  Q +  , o ) is abelian.           Closure property :                 Let a , b 𝞊  Q +       i.e a and b are positive rational numbers.                                            ⇒ ab/3 is also a positive rational number     ...

  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...

                                          Theorem : In a group , the identity element is unique :   proof :               Suppose G is a group and e be the identity element in G.              Now we prove e is only the identity element in G.              In contrary assume e' is another identity element in G.             To prove the identity element is unique, we have to prove e = e'.             Since e is identity in G, we have e a = a = a e ∀ a 𝞊 G.             In particular , since e' 𝞊 G, we have e e' = e' = e' e     .............I             Since e' is identity in G, we have e' a = a = a e' ∀ a 𝞊 G.     ...

Theorem : In a group G, inverse of  any element is unique.    Proof :              Let G be a group with the identity element e.                             Let a 𝞊 G.                         Since G is a group, a has an inverse element say b.                        Now we prove b is only the inverse element of a in G.                         In contrary, assume c is also the inverse of a in G.              To prove a has unique inverse element , we have to prove b = c.                            Since b is inverse of a in G, we have a b = e = b a ............I     ...

  Problem   :   Show that the operation ‘ o ‘ given by aob = a b is a binary operation on the set of natural numbers N. Is this operation associative and commutative in N? Solution :    Consider N,the set of all natural numbers and ‘o’ is operation defined on N such that aob = a b   ∀ a,b 𝟄 N.      Let a,b 𝟄 N.    Now aob = a b                    = a x a x … x b times                    = a natural number                    𝟄 N   ∴ aob 𝟄 N     ∀   a,b 𝟄 N   'o' is a binary operation on N.   Checking for associative law :   Let a,b,c 𝟄 N   Now ao(boc) = aob c       | since ...