in Set Theory & Algebra
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shouldnt the answer be 1??

in Set Theory & Algebra
329 views

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X = 2 n-1

Y = 2n

X /Y = 1/2

 ???

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ITS CORRECT 

but why X= 2n-1

why not 2n

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Gate Fever   got it ??

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1 Answer

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Best answer

take a small example and then analyze

1) --->when N  =  3 (Odd number)

          let S = { 1,2,3 }  --- > 8 subset possible of set S

X is subset of set of size 3 containing even number of elements

          X = { {$\Phi$ }  , {1,2} , {2,3} , {1,3}  }  = 4 subsets possible containing even number of elements

 2)  -->when N = 2 (even number)

                           S = {1,2} --- > 4 subset possible of set S

                           X = { {$\Phi$ } , {1,2} }  = 2 subsets possible  containing even number of elements

 

therefore the general formula is that if we have a sets of size = N (odd /even)

then the number of subsets with even cardinality equals the number of subsets with odd cardinality. So

this number is $\frac{1}{2}$2n=2n−1.

 

X =  2n−1

A = N elements

B  = {0 ,1 } -- > 2 elements

 

Total number of function possible from set A to set B = Y = 2n

 

$\frac{X}{Y} = \frac{2^{n-1}}{2^{n}} = \frac{1}{2}$

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thanks, got it!
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