Answers to logical reasoning exercises, showing converse, inverse, and contrapositive statements with highlighted text for clarity.

It looks like you've uploaded an image containing a list of conditional statements (if-then statements) along with their converse, inverse, and contrapositive forms. The task appears to be about understanding logical relationships in conditional statements.

Let’s go through the logic behind each and verify whether the answers provided are correct, then explain how to form these related statements properly.

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🔹 Key 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

> ✔ The contrapositive is logically equivalent to the original statement.
> ✘ The converse and inverse are not logically equivalent to the original.

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Now let's analyze each problem one by one:

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

- Original: If P, then Q
- P = You are in love
- Q = You are inspired

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

✔ All correct.

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

- P = Lines are parallel
- Q = Lines do not intersect

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

⚠️ Note: This assumes we're working in Euclidean geometry where parallel lines never meet — so this logic holds.

✔ All correct.

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

This is true because:
- A number divisible by both 2 and 3 must be divisible by LCM(2,3) = 6.

P = Divisible by 2 and 3
Q = Divisible by 6

| Form | Statement |
|------|----------|
| Converse | If a number is divisible by 6, then it is divisible by 2 and 3 → ✔ True and correct |
| Inverse | If a number is not divisible by 2 and 3, then it is not divisible by 6 → ⚠️ Ambiguous wording |

Wait! Let's fix this.

The original condition is: "divisible by 2 and 3" → so both must be true.

So “not divisible by 2 and 3” means not divisible by both — i.e., fails at least one.

But the inverse should be:
If not (divisible by 2 and 3), then not divisible by 6
→ Which is: If not (P), then not Q

So:
“If the number is not divisible by 2 or not divisible by 3, then it is not divisible by 6”

But the answer says:
> "If the number is not divisible by 2 and 3, then the number is not divisible by 6."

This is incorrectly worded — it implies both conditions fail, but the inverse should negate the entire antecedent.

Correct inverse:
> If the number is not divisible by 2 or not divisible by 3, then it is not divisible by 6.

But the given version says "not divisible by 2 and 3", which is ambiguous.

✘ So the inverse is poorly phrased and technically incorrect.

Contrapositive:
"If not divisible by 6, then not divisible by 2 and 3" → This is also problematic.

Actually, contrapositive should be:
"If not divisible by 6, then not (divisible by 2 and 3)"
→ i.e., not divisible by 2 or not divisible by 3

But the answer says:
> "If the number is not divisible by 6, then the number is not divisible by 2 and 3"

Again, this suggests both fail, which is too strong.

✘ So both inverse and contrapositive are incorrectly stated.

✔ Only the converse is correctly written.

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

Note: "garbage are" → should be "garbage is" (grammar issue, but we’ll ignore that).

P = Garbage is disposed properly
Q = Diseases are prevented

| Form | Statement |
|------|----------|
| Converse | If diseases were prevented, then garbage were disposed properly → ✔ Correct |
| Inverse | If garbage are not disposed properly, then diseases will not be prevented → ✔ Correct |
| Contrapositive | If diseases were not prevented, then garbage were not disposed properly → ✔ Correct |

✔ All correct.

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

P = Participate
Q = Get extra points

| Form | Statement |
|------|----------|
| Converse | If you get extra points, then you participated → ✔ Correct |
| Inverse | If you do not participate, then you will not get extra points → ✔ Correct |
| Contrapositive | If you do not get extra points, then you did not participate → ✔ Correct |

✔ All correct.

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

P = Moon is full
Q = Vampires are prowling

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

✔ All correct.

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

This is:
P = Not understand Geometry
Q = Not know deductive reasoning

So:
If P, then Q

Now:

| Form | Statement |
|------|----------|
| Converse | If you do not know how to reason deductively, then you do not understand Geometry → ✔ Correct |
| Inverse | If you understand Geometry, then you know how to reason deductively → ✔ Correct |
| Contrapositive | If you know how to reason deductively, then you understand Geometry → ✔ Correct |

✔ All correct.

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

P = No sense of balance
Q = Cannot skateboard

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

✔ All correct.

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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 is raining → ✔ Correct |
| Inverse | If it does not rain, then I go fishing → ✔ Correct |
| Contrapositive | If I go fishing, then it is not raining → ✔ Correct |

✔ All correct.

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

P = Read carefully
Q = Solution is easy

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

✔ All correct.

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🟩 Summary of Errors:

Only Problem #3 has issues due to logical negation of compound statements.

#### Problem 3: Fix Required

Original:
If divisible by 2 and 3, then divisible by 6

- Converse: If divisible by 6, then divisible by 2 and 3 → ✔ Correct
- Inverse: Should be:
> If not divisible by 2 or not divisible by 3, then not divisible by 6
But written as:
> "If not divisible by 2 and 3..." → ✘ Incorrect (suggests both fail)

- Contrapositive: Should be:
> If not divisible by 6, then not (divisible by 2 and 3) → i.e., not divisible by 2 or not divisible by 3
But written as:
> "If not divisible by 6, then not divisible by 2 and 3" → ✘ Incorrect

✔ So only #3 has flawed logic due to improper handling of conjunctions.

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✔ Final Answer:

All answers are correct except for problem #3, where the inverse and contrapositive are incorrectly worded due to misapplication of negation over "and".

#### Corrected Version for #3:

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

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💡 Tips for Students:

- Always remember:
- Negation of "P and Q" is "not P or not Q"
- Negation of "P or Q" is "not P and not Q"
- Use parentheses when dealing with compound statements.
- The contrapositive is always logically equivalent; others are not.

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