Student's handwritten answers on a logic and proof homework worksheet.

Let's go through the homework step by step and verify or correct each answer, explaining the logic behind conditional statements.

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Section 1: Identify the hypothesis and conclusion

#### 1. If the product of two numbers is 0, then at least one of the numbers must be 0.

- Hypothesis: *The product of two numbers is 0*
✔ Correct — this is the "if" part.
- Conclusion: *At least one of the numbers must be 0*
✔ Correct — this is the "then" part.

✔ Correctly identified.

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#### 2. If it is daylight saving time, then I must reset my clocks.

- Hypothesis: *If it is daylight saving time*
✔ Correct — this is the condition.
- Conclusion: *I must reset my clocks*
✔ Correct — this is the result.

✔ Correctly identified.

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Section 2: Write in if-then form

#### 3. A rhombus is a quadrilateral with four congruent sides.

Rewritten in if-then form:
👉 *If a quadrilateral is a rhombus, then it has four congruent sides.*

⚠️ Student’s answer: *"If a rhombus is a quadrilateral then it has four congruent sides."*

✘ Incorrect reasoning — The student reversed the logic. The original statement defines a rhombus, so the correct if-then form should start with the shape (rhombus) as the subject.

✔ Correct Answer:
> If a quadrilateral is a rhombus, then it has four congruent sides.

> (Alternatively: *If a figure is a rhombus, then it has four congruent sides.*)

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#### 4. Those that finish the marathon will get a medal.

Rewritten in if-then form:
👉 *If you finish the marathon, then you will get a medal.*

✔ Student's answer: *"If you finish the marathon then you will get a medal"*
✔️ Correct!

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#### 5. All freshmen are required to attend orientation.

This is a universal statement. To write it in if-then form:

👉 *If you are a freshman, then you must attend orientation.*

✔ Student wrote: *"If you are a freshman, you must attend orientation."*
✔️ This is acceptable, though slightly informal. It's essentially correct.

Better formal version:
> If a student is a freshman, then the student must attend orientation.

But the student's version is acceptable for this context.

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Section 3: Inverse, Converse, Contrapositive, Bi-conditional

Given statement:
"If you live in Dallas, then you live in Texas."

Let’s define:
- P: You live in Dallas
- Q: You live in Texas

So: P → Q

Now let’s analyze each:

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#### Inverse: Negate both hypothesis and conclusion
👉 *If you do not live in Dallas, then you do not live in Texas.*

- Truth value: ✘ False
- Counterexample: Someone can live in Houston (not Dallas), but still live in Texas.
- So, not living in Dallas doesn’t mean you don’t live in Texas.
- Thus, inverse is false.

✔ Student said: *False* — ✔️ Correct.

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#### Converse: Switch hypothesis and conclusion
👉 *If you live in Texas, then you live in Dallas.*

- Truth value: ✘ False
- Not everyone in Texas lives in Dallas (e.g., Austin, San Antonio).
- So, living in Texas does not imply living in Dallas.
- Thus, converse is false.

✔ Student said: *False* — ✔️ Correct.

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#### Contrapositive: Negate and switch (logically equivalent to original)
👉 *If you do not live in Texas, then you do not live in Dallas.*

- Truth value: ✔ True
- Because Dallas is in Texas — if you're not in Texas, you can't be in Dallas.
- Logically equivalent to original statement.

✔ Student said: *True* — ✔️ Correct.

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#### Bi-conditional: P ↔ Q
👉 *You live in Dallas if and only if you live in Texas.*

- This means:
- If you live in Dallas → you live in Texas (true)
- If you live in Texas → you live in Dallas (false)

Since the converse is false, the bi-conditional is false.

✔ Student said: *True* — ✘ Incorrect

➡️ Error here: The bi-conditional is only true if both the original and converse are true.

Since the converse ("If you live in Texas, then you live in Dallas") is false, the bi-conditional is false.

✔ Correct answer: False

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✔ Final Summary of Corrections:

| Question | Student Answer | Correct? | Correction |
|--------|----------------|---------|-----------|
| 3 | "If a rhombus is a quadrilateral..." | ✘ | Should be: "If a quadrilateral is a rhombus, then it has four congruent sides." |
| 5 | "If you are a freshman, you must attend orientation." | ✔ | Acceptable, though could be more formal |
| 6. Bi-conditional truth value | True | ✘ | Should be False |

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✔ Final Answer Key:

#### 3.
Correct if-then form:
> *If a quadrilateral is a rhombus, then it has four congruent sides.*

#### 5.
If-then form:
> *If you are a freshman, then you must attend orientation.*

#### 6. Bi-conditional:
> *If you live in Dallas, then you live in Texas.*
> Bi-conditional: *You live in Dallas if and only if you live in Texas.*
> Truth Value: ✘ False (because not all Texans live in Dallas)

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📝 Explanation:

- The bi-conditional (P ↔ Q) is true only when both P → Q and Q → P are true.
- Here, P → Q is true (Dallas ⊆ Texas), but Q → P is false (Texas ≠ Dallas).
- Therefore, P ↔ Q is false.

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✔ Final Answer:
The bi-conditional is FALSE, not TRUE. That is the main error on the page.

All other answers are correct except for #3 and #6.