Answer:
The given options seem to be related to properties of the subsets \( P \) and \( Q \) in relation to each other. Here's a clearer version of the options and what they imply:
Let \( P \) and \( Q \) be two non-empty subsets of a set \( R \) such that \( P \cap Q \neq \emptyset \). Then:
a) \( P \cap Q \) are disjoint sets.
b) \( Q \subseteq P \).
c) \( P \cap Q \) are non-disjoint sets.
d) \( P \subseteq Q \).
Given the condition \( P \cap Q \neq \emptyset \):
1. **Option (a) \( P \cap Q \) are disjoint sets:**
- This cannot be true because disjoint sets have an empty intersection, but it is given that \( P \cap Q \neq \emptyset \).
2. **Option (b) \( Q \subseteq P \):**
- This means that every element of \( Q \) is also an element of \( P \). This is possible but not necessarily true from the given condition.
3. **Option (c) \( P \cap Q \) are non-disjoint sets:**
- This is true by definition since \( P \cap Q \neq \emptyset \). Therefore, \( P \) and \( Q \) are not disjoint.
4. **Option (d) \( P \subseteq Q \):**
- This means that every element of \( P \) is also an element of \( Q \). This is possible but not necessarily true from the given condition.
Thus, the correct option is:
**c) \( P \cap Q \) are non-disjoint sets.**
Step-by-step explanation:
