If E and F are events with `P(E)leP(F)` and `P(EnnF)gt0`, then

Options

A. occurrence of E implies occurrence of F
B. occurrence of F implies occurrence of E
C. non-occurrence of E implies non-occurrence of F
D. None of the above implications holds

Answer

To solve the problem, we need to analyze the given conditions and determine the implications of events E and F based on their probabilities and intersections.

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
- We know that \( P(E) \leq P(F) \) which means the probability of event E is less than or equal to the probability of event F.
- We also know that \( P(E \cap F) > 0 \) which indicates that events E and F have a non-empty intersection; they can occur simultaneously.

2. **Visualizing with a Venn Diagram**:
- Draw a Venn diagram with two circles, one for event E and one for event F.
- Since \( P(E) \leq P(F) \), the area representing event E should be smaller than or equal to the area representing event F.

3. **Analyzing the Options**:
- **Option 1**: "Occurrence of E implies occurrence of F".
- If E occurs, then F must also occur. However, since E can occur in a part of the circle that does not overlap with F, this option is incorrect.

- **Option 2**: "Occurrence of F implies occurrence of E".
- If F occurs, then E must also occur. This is also incorrect since F can occur independently of E in the non-overlapping area of F.

- **Option 3**: "Non-occurrence of E implies non-occurrence of F".
- If E does not occur, it does not necessarily mean that F does not occur. F can occur even if E does not, hence this option is incorrect as well.

- **Option 4**: "None of the above implications holds".
- Since all previous options are incorrect, this option must be correct.

4. **Conclusion**:
- Based on the analysis of the options, we conclude that the correct answer is option 4: "None of the above implications holds".

### Final Answer:
The correct option is 4: "None of the above implications holds".