devotion & mathematics

Small Set Vs Finite Set :          In mathematics, from so many years there is a confusion to few members who are interested  in  learning mathematics about small and and finite set. Most of people thought that small set and  finite set  are same. Basically small  and finite sets are not same.           We know a set is called finite set if the length of set is finite. Then  what about small set. In  mathematics, a small set does not have one universally fixed meaning. But with a small example,  we can found that what is a small set and how it differs from finite set.                   Example :                Consider the real line ℛ. We all know that  ℛ is an infinite set. Hence it is clear that  ℛ  is not a finite  set.  Even though ℛ is not a finite set , we can not conclude ...

  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...

    what heppened when   ∞/∞ =1?  " Infinity " , In my earlier post , we have discussed about infinity and its properties. Come to our question " what happened when ∞ / ∞ = 1 ? " Assume   ∞/∞ = 1 that implies ∞ = ∞. Let the ∞ in the L.H.S. is the number of water drops in a riverand the ∞ in the R.H.S. is the  number of  water drops in a sea. It is impossible to count the number of water drops in a sea as  well as in the river.  That why we  consider both having infinite water drops. Now as our  assumption  if ∞ = ∞ then the  number of water drops in a river must equal to the number of  water  drops in a sea it results  the river  and sea must be of equal in size.  Is it true? No. Hence ∞  ≠ ∞ i.e. ∞/∞ ≠ 1. First of all this question may arise only if ∞ is a number.  Now my question " Is ∞ a mumber ? "  Answer is  ∞ is not a number. It is a symbolic representation of n...

                                          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 ...