Test by Bikram | Mathematics | Test 2 | Question: 30

Question

Let $S$ be the set $\{1,2,3,  \dots ,8\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $S$. The value of $n$ is _______.

Comments

Similar to this problem : https://gateoverflow.in/94195/probability-settheory

Answer 1

(3^8 - 1)/2 - 2^8 + 1 [from matrix diagram and counting only off diagonal 1's and leaving 1's related to phi set]

Comments

Sir, please elaborate the solution.

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see this problem  https://gateoverflow.in/94195/probability-settheory , how it solved and try yourself .

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@Bikram Sir in answer why are you adding 1 related to phi set as in question it is mentioned that two sets are non empty.

Answer 2

Lets suppose two sets be A and B.

Elements can be either in A or B or not in both sets.

Therefore, Total possible ordered pairs = $(3)^8$

But cases like,

1. A={ø} and B = elements
2. A= elements and B= {ø}

are not allowed, remove them.

So, $3^8$ – $2 *2^8$

But we have subtracted A={ø} and B={ø} two times, so add $1$.

Current total = $3^8$ – $2 * 2 ^ 8$ + $1$ = $ 6050 $

But here we have to consider unordered pairs, so resultant total =  $6050/2$ =$3025$.