Question

Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S equal to,
$\left(a\right)$ 25
$\left(b\right)$ 34
$\left(c\right)$ 42
$\left(d\right)$ 41

Answer 1

Hint: In this particular question use the concept that total number of unordered pairs of disjoint subsets of any set containing n elements is equal to ${\displaystyle \frac{3^n+1}{2}}$, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
S = {1, 2, 3, 4}
Now we have to find out the total number of unordered pairs of disjoint subsets of S.
Disjoint sets – Two sets are said to be disjoint sets if they have no elements in common i.e. the intersection of these sets is an empty set.
For example: A = {1, 2, 3} and B = {4, 5, 6}, so A, and B are called Disjoint sets.
Now we have to find out the total number of subsets of the given set S, such that the subsets are unordered pairs of disjoint subsets of S.
 Now as we know that the total number of unordered pairs of disjoint subsets of any set containing n elements is equal to, N = ${\displaystyle \frac{3^n+1}{2}}$.................... (1), where n = 1, 2, 3......
Now in the given subset number of elements is equal to 4, so n = 4.
Now substitute this value in equation (1) we have,
Therefore, N = ${\displaystyle \frac{3^4+1}{2}}$
Now simplify this we have,
$\Rightarrow N={\displaystyle \frac{81+1}{2}}={\displaystyle \frac{82}{2}}=41$
So there are 41 unordered pairs of disjoint subsets of S.
So this is the required answer.
Hence option (d) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall that two sets are said to be a disjoint sets if they have no elements in common i.e. the intersection of these sets is an empty set, and always recall the formula to find out the total number of disjoint sets if a set contain n number of elements which is stated above.