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Let's go through the problem step by step.

### Part a: True/False Questions
The sample space is \( S = \{1, 2, 3, 4, 5, 6\} \), with two sets \( A = \{1, 4, 7\} \) and \( B = \{2, 3, 5\} \).

i. \( A \subseteq S \): 
   - False. Set \( A \) contains the element 7, which is not in the sample space \( S \). So \( A \) is **not** a subset of \( S \).

ii. \( 4 \in B \):
   - False. The element 4 is not in set \( B \). So this statement is **false**.

iii. \( n(B) = 3 \):
   - True. Set \( B \) has three elements: \( 2, 3, 5 \). So \( n(B) = 3 \).

iv. \( A \cap B = \emptyset \): 
   - True. The intersection of \( A \) and \( B \) is the empty set because they have no common elements. So this statement is **true**.

v. \( A \cup B = \emptyset \): 
   - False. The union of \( A \) and \( B \) would include all the elements in both sets, i.e., \( A \cup B = \{1, 2, 3, 4, 5, 7\} \), which is not empty. So this statement is **false**.

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### Part b: Finding probabilities
The sample space \( S \) has 6 elements: \( \{1, 2, 3, 4, 5, 6\} \).

i. \( P(B) \):
   - The probability of \( B \) is the ratio of the number of elements in \( B \) to the number of elements in the sample space \( S \).
   \[
   P(B) = \frac{|B|}{|S|} = \frac{3}{6} = \frac{1}{2}
   \]

ii. \( P(B^c) \) (the complement of \( B \)):
   - The complement of \( B \) is the set of elements in the sample space that are not in \( B \), i.e., \( B^c = \{1, 4, 6\} \). The probability is:
   \[
   P(B^c) = \frac{|B^c|}{|S|} = \frac{3}{6} = \frac{1}{2}
   \]

iii. \( P(A \cap B) \):
   - From part a.iv, \( A \cap B = \emptyset \), meaning the two sets have no common elements. Thus, the probability of their intersection is:
   \[
   P(A \cap B) = 0
   \]

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Would you like further clarification or details on any part of the solution?

Here are some related questions to extend the understanding:

1. What is the union \( A \cup B \) in this case, and how would we calculate its probability?
2. How would we calculate the probability of the complement of \( A \)?
3. If set \( C = \{2, 6\} \), what would be \( P(C) \)?
4. How would the answer change if the sample space included 7 as well?
5. What does it mean if two sets are mutually exclusive in terms of probability?

**Tip:** The complement of a set \( B \) in probability refers to everything in the sample space that is not in \( B \). So, \( P(B^c) = 1 - P(B) \).

For the sample space {1, 2, 3, 4, 5, 6} and the sets A = {1, 4, 7} and B = {2, 3, 5}, answer the following: a) State whether the statements are true or false; b) Find the probabilities.

Explore a probability problem involving sets A and B from a sample space {1, 2, 3, 4, 5, 6}. Determine the truth of set-related statements and calculate probabilities. Concepts include the complement rule, set intersections, and unions.

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