When a die is rolled, sample space S = {1, 2, 3, 4, 5, 6}. Let some of the events are A = {2, 3}, B = {1, 3,5}, C = {4, 6}, D = {6} and E = {1,5}.
(ii) Events A and Care Mutually Exclusive Events.

Answer

To determine whether events A and C are mutually exclusive, we follow these steps:

### Step 1: Identify the Sample Space and Events
The sample space \( S \) when a die is rolled is:
\[ S = \{1, 2, 3, 4, 5, 6\} \]

The events are defined as:
- \( A = \{2, 3\} \)
- \( C = \{4, 6\} \)

### Step 2: Find the Intersection of Events A and C
To check if events A and C are mutually exclusive, we need to find the intersection of these two events, denoted as \( A \cap C \).

The intersection \( A \cap C \) consists of all elements that are common to both sets A and C.

### Step 3: Calculate the Intersection
- Elements in \( A \) are \( 2 \) and \( 3 \).
- Elements in \( C \) are \( 4 \) and \( 6 \).

Now, we find the common elements:
\[ A \cap C = \{2, 3\} \cap \{4, 6\} = \emptyset \]

### Step 4: Conclusion
Since the intersection \( A \cap C \) is empty (i.e., \( A \cap C = \emptyset \)), it means that events A and C do not share any outcomes. Therefore, they are mutually exclusive events.

### Final Answer
Yes, events A and C are mutually exclusive.

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