If E and F are events with P(E) `le` P(F) and P(E `nn` F) `gt` 0, then

Options

A. occurrence of E `rArr` occurrence of F
B. occurrence of F `rArr` occurrence of E
C. non-occurrence of E `rArr` non-occurrence of F
D. none of the above implication hold

Answer

To solve the problem, we need to analyze the given conditions regarding the events E and F.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have two events, E and F.
- It is given that \( P(E) \leq P(F) \).
- It is also given that \( P(E \cap F) > 0 \).

2. **Interpreting the Probabilities**:
- The condition \( P(E) \leq P(F) \) indicates that the probability of event E occurring is less than or equal to the probability of event F occurring.
- The condition \( P(E \cap F) > 0 \) means that there is a non-zero probability that both events E and F occur simultaneously.

3. **Visualizing with a Venn Diagram**:
- We can represent the events E and F in a Venn diagram.
- The area representing E will be smaller than or equal to the area representing F, and there will be an overlap between E and F since \( P(E \cap F) > 0 \).

4. **Analyzing the Options**:
- We need to evaluate the implications of the given conditions based on the options provided:
1. **Option 1**: Occurrence of E implies occurrence of F.
- This is incorrect because E can occur without F occurring.
2. **Option 2**: Occurrence of F implies occurrence of E.
- This is also incorrect because F can occur without E occurring.
3. **Option 3**: Non-occurrence of E implies non-occurrence of F.
- This is incorrect as well; F can occur even if E does not.
4. **Option 4**: None of the above implications hold.
- This is the correct option since all previous options are false.

5. **Conclusion**:
- Therefore, the correct answer is that none of the implications hold true based on the given conditions.

### Final Answer:
The correct option is **4. None of the above implications hold.**