Practice exam worksheet focused on identifying the converse, inverse, and contrapositive of logical implications.

Let's solve this step by step. The task is to find the converse, inverse, and contrapositive of each given conditional statement.

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🔷 Definitions:

For a conditional statement:
If P, then Q
- Converse: If Q, then P
- Inverse: If not P, then not Q
- Contrapositive: If not Q, then not P

> Note: The contrapositive is logically equivalent to the original statement.
> The converse and inverse are logically equivalent to each other but not necessarily to the original.

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Now let’s go through each problem one by one.

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1. If you are in love, then you are inspired.

- P: You are in love
- Q: You are inspired

| Form | Statement |
|------|-----------|
| Converse | If you are inspired, then you are in love. |
| Inverse | If you are not in love, then you are not inspired. |
| Contrapositive | If you are not inspired, then you are not in love. |

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2. If two lines are parallel, then the two lines do not intersect.

- P: Two lines are parallel
- Q: The two lines do not intersect

| Form | Statement |
|------|-----------|
| Converse | If two lines do not intersect, then they are parallel. |
| Inverse | If two lines are not parallel, then they intersect. |
| Contrapositive | If two lines intersect, then they are not parallel. |

> ⚠️ Note: This is true in Euclidean geometry, but the converse may not always be true (e.g., skew lines in 3D don’t intersect but aren’t parallel). But we're only rewriting logically.

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3. If the number is divisible by 2 and 3, then the number is divisible by 6.

- P: The number is divisible by 2 and 3
- Q: The number is divisible by 6

| Form | Statement |
|------|-----------|
| Converse | If a number is divisible by 6, then it is divisible by 2 and 3. |
| Inverse | If a number is not divisible by 2 or not divisible by 3, then it is not divisible by 6. |
| Contrapositive | If a number is not divisible by 6, then it is not divisible by both 2 and 3. |

> ✔ This contrapositive is logically equivalent and correct.

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4. If garbage are disposed properly, then the diseases will be prevented.

- P: Garbage are disposed properly
- Q: Diseases will be prevented

Note: "Garbage are" → should be "garbage is" for grammar, but we’ll keep as is.

| Form | Statement |
|------|-----------|
| Converse | If diseases will be prevented, then garbage are disposed properly. |
| Inverse | If garbage are not disposed properly, then diseases will not be prevented. |
| Contrapositive | If diseases are not prevented, then garbage are not disposed properly. |

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5. If you participate in class, then you will get extra points.

- P: You participate in class
- Q: You will get extra points

| Form | Statement |
|------|-----------|
| Converse | If you get extra points, then you participated in class. |
| Inverse | If you do not participate in class, then you will not get extra points. |
| Contrapositive | If you do not get extra points, then you did not participate in class. |

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6. If the moon is full, then the vampires are prowling.

- P: The moon is full
- Q: Vampires are prowling

| Form | Statement |
|------|-----------|
| Converse | If the vampires are prowling, then the moon is full. |
| Inverse | If the moon is not full, then the vampires are not prowling. |
| Contrapositive | If the vampires are not prowling, then the moon is not full. |

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7. If you do not understand geometry, then you do not know how to reason deductively.

- P: You do not understand geometry
- Q: You do not know how to reason deductively

We can rephrase:
- Original: If ¬G, then ¬R
(where G = understand geometry, R = reason deductively)

So:
- Converse: If you do not know how to reason deductively, then you do not understand geometry.
(If ¬R, then ¬G)
- Inverse: If you understand geometry, then you know how to reason deductively.
(If G, then R) — this is the contrapositive of the original!
- Contrapositive: If you know how to reason deductively, then you understand geometry.
(If R, then G)

Wait! Let's clarify:

Original: If ¬G → ¬R
- Contrapositive: If R → G
- Converse: If ¬R → ¬G
- Inverse: If G → R

So:

| Form | Statement |
|------|-----------|
| Converse | If you do not know how to reason deductively, then you do not understand geometry. |
| Inverse | If you understand geometry, then you know how to reason deductively. |
| Contrapositive | If you know how to reason deductively, then you understand geometry. |

✔ All correct.

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8. If you do not have a sense of balance, then you cannot skateboard.

- P: You do not have a sense of balance
- Q: You cannot skateboard

| Form | Statement |
|------|-----------|
| Converse | If you cannot skateboard, then you do not have a sense of balance. |
| Inverse | If you have a sense of balance, then you can skateboard. |
| Contrapositive | If you can skateboard, then you have a sense of balance. |

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9. If it rains, then I do not go fishing.

- P: It rains
- Q: I do not go fishing

| Form | Statement |
|------|-----------|
| Converse | If I do not go fishing, then it rains. |
| Inverse | If it does not rain, then I go fishing. |
| Contrapositive | If I go fishing, then it does not rain. |

> Note: This makes logical sense — if you *do* go fishing, it must not be raining.

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10. If you read the question carefully, then the solution is easy.

- P: You read the question carefully
- Q: The solution is easy

| Form | Statement |
|------|-----------|
| Converse | If the solution is easy, then you read the question carefully. |
| Inverse | If you do not read the question carefully, then the solution is not easy. |
| Contrapositive | If the solution is not easy, then you did not read the question carefully. |

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✔ Final Answers:

Here is the completed table:

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#### 1. If you are in love, then you are inspired
- Converse: If you are inspired, then you are in love.
- Inverse: If you are not in love, then you are not inspired.
- Contrapositive: If you are not inspired, then you are not in love.

#### 2. If two lines are parallel, then the two lines do not intersect
- Converse: If two lines do not intersect, then they are parallel.
- Inverse: If two lines are not parallel, then they intersect.
- Contrapositive: If two lines intersect, then they are not parallel.

#### 3. If the number is divisible by 2 and 3, then the number is divisible by 6
- Converse: If a number is divisible by 6, then it is divisible by 2 and 3.
- Inverse: If a number is not divisible by 2 or not divisible by 3, then it is not divisible by 6.
- Contrapositive: If a number is not divisible by 6, then it is not divisible by both 2 and 3.

#### 4. If garbage are disposed properly, then the diseases will be prevented
- Converse: If diseases will be prevented, then garbage are disposed properly.
- Inverse: If garbage are not disposed properly, then diseases will not be prevented.
- Contrapositive: If diseases are not prevented, then garbage are not disposed properly.

#### 5. If you participate in class, then you will get extra points
- Converse: If you get extra points, then you participated in class.
- Inverse: If you do not participate in class, then you will not get extra points.
- Contrapositive: If you do not get extra points, then you did not participate in class.

#### 6. If the moon is full, then the vampires are prowling
- Converse: If the vampires are prowling, then the moon is full.
- Inverse: If the moon is not full, then the vampires are not prowling.
- Contrapositive: If the vampires are not prowling, then the moon is not full.

#### 7. If you do not understand geometry, then you do not know how to reason deductively
- Converse: If you do not know how to reason deductively, then you do not understand geometry.
- Inverse: If you understand geometry, then you know how to reason deductively.
- Contrapositive: If you know how to reason deductively, then you understand geometry.

#### 8. If you do not have a sense of balance, then you cannot skateboard
- Converse: If you cannot skateboard, then you do not have a sense of balance.
- Inverse: If you have a sense of balance, then you can skateboard.
- Contrapositive: If you can skateboard, then you have a sense of balance.

#### 9. If it rains, then I do not go fishing
- Converse: If I do not go fishing, then it rains.
- Inverse: If it does not rain, then I go fishing.
- Contrapositive: If I go fishing, then it does not rain.

#### 10. If you read the question carefully, then the solution is easy
- Converse: If the solution is easy, then you read the question carefully.
- Inverse: If you do not read the question carefully, then the solution is not easy.
- Contrapositive: If the solution is not easy, then you did not read the question carefully.

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✔ Summary:

This exercise helps students understand how to manipulate conditional statements logically. Remember:
- Contrapositive is logically equivalent to the original.
- Converse and Inverse are not necessarily true even if the original is.

Let me know if you'd like this formatted as a printable answer sheet!