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what happened when ∞/∞ = 1 ? : Real Analysis

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    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...
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Uniqueness of identity in a group : Group theory

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                                          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.     ...
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Uniqueness of inverse element : Group Theory

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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     ...
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Is aob = a^b is a binary operation on N? : Group Theory

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  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 ...
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Is aob = a is associative in S ? : Group Theory

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  Problem :   Let S be a non-empty set and ‘ o ‘ be an operation on S defined by aob = a for a,b 𝟄 S. Determine whether   o is commutative and associative in S? Solution :   Let S be a non-empty set and ‘ o ‘ be an operation on S defined by aob = a for a,b 𝟄 S. Checking for associative :     Let a,b,c 𝟄 S.    Now (aob)oc = aoc         | since aob = a |                            =   a              | since aoc = a |       ∴ (aob)oc = a   Also   ao(boc) = aob           | since boc = b |                        ...
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Basic Definitions : Group Theory #group theory

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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...
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Euler's summation formula : Number Theory #Euler's #summation #formula #: #Number #Theory

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Euler's summation formula : Statement :        If f(x) has continuous derivative f’ on [a,b] where 0 < a< b , then          𝜮 a<n ≤ b    f(n) = ∫ f(t)dt + ∫ (t – [t] ) dt +f(b)( [b] – b) -f(a) ( [a] -a ).   Proof :    Suppose f(x) has continuous derivative and f’(x) is in the closed interval [a,b] where 0 < a < b.   Let [a] =m and   [b] = k.   Then 𝜮 a<n ≤ b    f(n) = 𝜮 m<n ≤ k    f(n) = 𝜮 f(n)   …(i)            | where in third sigma n is from m+1 to k |   Suppose (n-1) and n are two integers in [a,b] and t lies between (n-1) and n. Then ∫ n-1 n [t] f’(t) dt = ∫ (n-1) f’(t) dt                                ...
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