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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pmFollowed by: 4 members; Thanked: 29 times

by vipulgoyal » Wed Mar 20, 2013 3:34 am

A set of n elements has 2^n subsets, including both the set itself and the empty set. How many common subsets do set {1, 2, 3, 4} and set {2, 3, 4, 5} have?

Ans not given, dont want to make all subsets and find any alt approach will be welcomed

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Source: Beat The GMAT — Problem Solving | Wed Mar 20, 2013 3:34 am

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Anju@Gurome: GMAT Instructor; Posts: 511; Joined: Wed Aug 11, 2010 9:47 amFollowed by: 86 members; Location: Delhi, India; Thanked: 344 times

by Anju@Gurome » Wed Mar 20, 2013 3:38 am

vipulgoyal wrote:A set of n elements has 2^n subsets, including both the set itself and the empty set. How many common subsets do set {1, 2, 3, 4} and set {2, 3, 4, 5} have?

The common subsets can only have the common elements : 2, 3, and 4.
Hence, number of common subsets = Number of subsets of the set {2, 3, 4} = 2^3 = 8

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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pmFollowed by: 4 members; Thanked: 29 times

by vipulgoyal » Wed Mar 20, 2013 4:17 am

Thanks, great logic

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rintoo22: Senior | Next Rank: 100 Posts; Posts: 73; Joined: Sun May 06, 2012 2:19 am; Location: Cape Town; Thanked: 6 times

by rintoo22 » Wed Mar 20, 2013 4:47 am

Hi Anju,

Great Logic. however I have a query that may enable me to understand the Subset theory.

I tried to jot down the subsets of each and somehow I am missing the subsets. As per the formula the 4 elements subset should have 16

{1, 2, 3, 4} :
{(1),(2),(3),(4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1, 2, 3, 4),()} = 12

Similarly
{2, 3, 4, 5}
{(2),(3),(4),(5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(2, 3, 4, 5),()} = 12

which subsets am I missing ?

Thanks
rintoo
(Ritesh)

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Anju@Gurome: GMAT Instructor; Posts: 511; Joined: Wed Aug 11, 2010 9:47 amFollowed by: 86 members; Location: Delhi, India; Thanked: 344 times

by Anju@Gurome » Wed Mar 20, 2013 4:50 am

rintoo22 wrote:which subsets am I missing ?

Subsets containing three elements.
In 1st case : {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, and {2, 3, 4}
In 2nd case : {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, and {3, 4, 5}

I'll take the opportunity to explain why the a set of n elements has 2^n subsets.

To create a subset, we may or may not select each of the n elements.
Hence, for each element, we have 2 options : either include it in the subset or don't.
So, total number of possible subsets = 2*2*2* ... (n times) = 2^n
When none of the elements are included in the subset, we get the null set and when all of the elements are included in the subset, we get the original set itself.

Hope that helps.

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