Let A, B and C be three events associated to a random experiment.
Statement-1 : If `A cap B sube C, " then " P(C )ge P(A)+P(B)-1`.
Statement-2 : If `P{(A cap B) cup (B cap C) cup (C cap A)} le " min" {P(A cup B),P(B cup C),P(C cup A)}`

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

To solve the given problem, we need to analyze the two statements provided regarding the events A, B, and C.

### Step-by-Step Solution:

**Step 1: Analyze Statement 1**
- The statement is: If \( A \cap B \subseteq C \), then \( P(C) \geq P(A) + P(B) - 1 \).
- This is a well-known result in probability theory, known as the inclusion-exclusion principle.
- To understand this, we can rewrite the condition \( A \cap B \subseteq C \) in terms of probabilities.
- If \( A \cap B \) is a subset of \( C \), then the probability of \( C \) must be at least as large as the probability of \( A \cap B \).
- Therefore, we can express it as:
\[ P(C) \geq P(A \cap B) \]
- According to the inclusion-exclusion principle, we have:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]
- Since \( P(A \cup B) \leq 1 \), we can derive:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]
- Thus, we conclude that \( P(C) \geq P(A) + P(B) - 1 \) holds true.

**Step 2: Analyze Statement 2**
- The statement is: If \( P((A \cap B) \cup (B \cap C) \cup (C \cap A)) \leq \min \{P(A \cup B), P(B \cup C), P(C \cup A)\} \).
- This statement is about the probabilities of unions and intersections of events.
- The left side represents the probability of the union of the intersections of the events.
- The right side represents the minimum of the probabilities of the unions of the pairs of events.
- If the left side is less than or equal to the minimum of the right side, it indicates that the overlaps among the events are not exceeding the individual unions of the pairs.
- This can be interpreted as a condition of independence or non-overlapping nature of events.
- Therefore, this statement is also true under the conditions described.

### Conclusion:
Both statements are true based on the principles of probability.