In an experiment of rolling of a fair die. Let A, B and C be three events defined as under:
A: a number which is a perfect square
B: a prime number
C: a number which is greater than 5.
Is A, B, and C exhaustive events?

Answer

To determine whether the events A, B, and C are exhaustive, we need to analyze each event based on the sample space of rolling a fair die.

### Step-by-Step Solution:

1. **Identify the Sample Space (S)**:
The sample space when rolling a fair die consists of the numbers:
\[ S = \{1, 2, 3, 4, 5, 6\} \]

2. **Define Event A**:
Event A is defined as the occurrence of a perfect square. The perfect squares in the sample space are:
\[ A = \{1, 4\} \]
(since \(1^2 = 1\) and \(2^2 = 4\)).

3. **Define Event B**:
Event B is defined as the occurrence of a prime number. The prime numbers in the sample space are:
\[ B = \{2, 3, 5\} \]
(since 2, 3, and 5 are the only prime numbers in the range of 1 to 6).

4. **Define Event C**:
Event C is defined as the occurrence of a number greater than 5. The only number in the sample space that is greater than 5 is:
\[ C = \{6\} \]

5. **Find the Union of Events A, B, and C**:
Now, we will find the union of these three events:
\[ A \cup B \cup C = \{1, 4\} \cup \{2, 3, 5\} \cup \{6\} \]
Combining these sets gives us:
\[ A \cup B \cup C = \{1, 2, 3, 4, 5, 6\} \]

6. **Check for Exhaustiveness**:
An exhaustive set of events covers the entire sample space. Since:
\[ A \cup B \cup C = S \]
we conclude that events A, B, and C together cover all possible outcomes when rolling a die.

### Conclusion:
Since the union of events A, B, and C equals the sample space, we can conclude that A, B, and C are exhaustive events.

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