Write the inverse converse ansd contrapositive of implications:
if I neglect my studies then I fail in the examination.

Answer

To solve the problem, we need to identify the original implication and then derive its inverse, converse, and contrapositive.

### Step-by-Step Solution:

1. **Identify the Original Implication**:
The original statement is: "If I neglect my studies, then I fail in the examination."
- Let \( P \): "I neglect my studies."
- Let \( Q \): "I fail in the examination."
- Therefore, the implication can be written as \( P \implies Q \).

2. **Find the Inverse**:
The inverse of an implication \( P \implies Q \) is given by \( \neg P \implies \neg Q \).
- Negation of \( P \): "I do not neglect my studies."
- Negation of \( Q \): "I do not fail in the examination."
- Thus, the inverse is: "If I do not neglect my studies, then I do not fail in the examination."

3. **Find the Converse**:
The converse of an implication \( P \implies Q \) is given by \( Q \implies P \).
- Therefore, the converse is: "If I fail in the examination, then I neglect my studies."

4. **Find the Contrapositive**:
The contrapositive of an implication \( P \implies Q \) is given by \( \neg Q \implies \neg P \).
- Negation of \( Q \): "I do not fail in the examination."
- Negation of \( P \): "I do not neglect my studies."
- Thus, the contrapositive is: "If I do not fail in the examination, then I do not neglect my studies."

### Final Results:
- **Original Implication**: If I neglect my studies, then I fail in the examination.
- **Inverse**: If I do not neglect my studies, then I do not fail in the examination.
- **Converse**: If I fail in the examination, then I neglect my studies.
- **Contrapositive**: If I do not fail in the examination, then I do not neglect my studies.