Question

Which of the following set is partition of the sample space we get while throwing a die?

• A. $\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6\left)\right\}$
• B. $\left\{\right(1),(2),(3),(4),(5),(6\left)\right\}$
• C. $\left\{\right(1,2),(3,4\left)\right\}$
• D. $\left\{\right(1,2,3,4),(5,6\left)\right\}$

Answer 1

Answer: (B), (D)

The correct options are
B $\left\{\right(1),(2),(3),(4),(5),(6\left)\right\}$
D $\left\{\right(1,2,3,4),(5,6\left)\right\}$

To solve this question, we need to know partition of a sample space S. A collection of sets is said to be a partition of a set S if the sets in the collection are disjoint and their union is S. So we will go through each option and check if it satisfies both the conditions.
1) Union gives sample space S = {1,2,3,4,5,6}
2) All of them are disjoint
Option A) Union of (1,2), (2,3), (3,4), (4,5) and (5,6) will give the sample space {1,2,3,4,5,6}. But there is a common element for (1,2) and (2,3). So these sets are not disjoint.
Option B) { (1), (2), (3), (4), (5), (6) }. Here, the union will give the sample and there are no common element for any of those pairs. So, we call this a partition of sample space
Option C) {(1,2), (1,2,3,4,5,6), } : This is not a partition because there are two common elements, 1 and 2
Option D) {(1, 2), (3,4)} : In this case the union won’t give the sample space. 5 and 6 will be missing from sample space.
$\Rightarrow$ not a partition of sample space
Option E) { (1, 2, 3, 4), (5, 6) }: No elements in common and the union gives the sample space.
$\Rightarrow$ Partition of sample space.