Counting

[vipulgoyal]

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

[Anju@Gurome]

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

[vipulgoyal]

Thanks, great logic

[rintoo22]

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)

[Anju@Gurome]

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.