Student worksheet on logic and proof, focusing on conditional statements and their components.

Let's go through the homework step by step and evaluate each part. The assignment is about conditional statements in logic, including identifying hypothesis and conclusion, rewriting statements in "if-then" form, and forming related logical statements (inverse, converse, contrapositive, biconditional), along with their truth values.

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Part 1: Identify Hypothesis and Conclusion

#### Problem 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* ✔
- Conclusion: *At least one of the numbers must be 0* ✔

✔️ Correct. This is a well-known property of multiplication (Zero Product Property).

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

- Hypothesis: *It is daylight saving time* ✔
- Conclusion: *I must reset my clocks* ✔

✔️ Correct. The "if" part is the hypothesis, the "then" part is the conclusion.

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Part 2: Write in "If-Then" Form

#### Problem 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.

(Alternatively: *If a quadrilateral has four congruent sides, then it is a rhombus.* — but this is not logically equivalent unless specified. However, the original statement says "a rhombus is...", so the correct conditional is that being a rhombus implies having four congruent sides.)

So, the student wrote:
> *If a rhombus is a quadrilateral, then it has four congruent sides.*

This is not quite correct because it’s redundant to say "if a rhombus is a quadrilateral"—all rhombuses are quadrilaterals. The key idea is that being a rhombus implies having four congruent sides.

✔ Better version:
If a quadrilateral is a rhombus, then it has four congruent sides.

But the student’s answer is close, though slightly awkward. It could be accepted with clarification.

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

Rewritten:
✔ If you finish the marathon, then you will get a medal.

✔️ Correct. This is a standard conversion from a general statement to conditional form.

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#### Problem 5:
> All freshmen are required to attend orientation.

Rewritten in if-then form:
✔ If you are a freshman, then you must attend orientation.

✔️ Correct. This captures the meaning accurately.

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Part 3: Inverse, Converse, Contrapositive, Biconditional

#### Original 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 analyze:

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

- Truth Value: ✘ False
- Why? Because someone can live in Houston (Texas) but not in Dallas. So, not living in Dallas doesn’t mean you don’t live in Texas.
- Counterexample: A person who lives in Austin lives in Texas but not in Dallas.

✔ Student wrote:
> If you do not live in Dallas, then you do not live in Texas.
> Truth Value: False ✔
✔️ Correct.

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

- Truth Value: ✘ False
- Why? Not everyone in Texas lives in Dallas. Many live in other cities.

✔ Student wrote:
> If you live in Texas, then you live in Dallas.
> Truth Value: False ✔
✔️ Correct.

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#### 3. Contrapositive:
Negate and switch:
> If you do not live in Texas, then you do not live in Dallas.

- Truth Value: ✔ True
- Why? Because Dallas is in Texas. If you're not in Texas, you can't be in Dallas. This is logically equivalent to the original.

✔ Student wrote:
> If you do not live in Texas, then you do not live in Dallas.
> Truth Value: True ✔
✔️ Correct.

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#### 4. Biconditional:
> If you live in Dallas, then you live in Texas, and if you live in Texas, then you live in Dallas.

This would be: P ↔ Q, which means “You live in Dallas if and only if you live in Texas.”

But this is false, because living in Texas does not imply living in Dallas.

So the biconditional is false.

✔ Student wrote:
> If you live in Dallas, then you live in Texas.
> Truth Value: True ✘

Wait! There's a problem here.

The biconditional should be written as:
> You live in Dallas if and only if you live in Texas.

Or:
> If you live in Dallas, then you live in Texas, and if you live in Texas, then you live in Dallas.

But the student only wrote the original statement, not the biconditional.

✘ Incorrect. The student failed to write the biconditional properly.

Correct Answer:
> If you live in Dallas, then you live in Texas, and if you live in Texas, then you live in Dallas.
> Or more simply: *You live in Dallas if and only if you live in Texas.*
> Truth Value: False (because the converse is false)

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✔ Final Summary of Corrections:

| Problem | Student's Answer | Correctness | Notes |
|--------|------------------|-------------|-------|
| 1 | Hypothesis: Product = 0; Conclusion: One number is 0 | ✔ Correct | Good |
| 2 | Hypothesis: DST; Conclusion: Reset clocks | ✔ Correct | Good |
| 3 | If a rhombus is a quadrilateral, then it has 4 congruent sides | ⚠️ Partially correct | Should be: *If a quadrilateral is a rhombus, then it has 4 congruent sides.* |
| 4 | If you finish the marathon, then you get a medal | ✔ Correct | Good |
| 5 | If you are a freshman, then you must attend orientation | ✔ Correct | Good |
| 6a. Inverse | If not in Dallas, then not in Texas | ✔ Correct | False – good |
| 6b. Converse | If in Texas, then in Dallas | ✔ Correct | False – good |
| 6c. Contrapositive | If not in Texas, then not in Dallas | ✔ Correct | True – good |
| 6d. Biconditional | Repeated original statement | ✘ Incorrect | Must include both directions; truth value should be False |

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✔ Final Answer:

The student did very well on most parts, especially understanding hypothesis/conclusion and inverse/converse/contrapositive. The only error is in #6d (Biconditional) — they did not write the full biconditional and incorrectly labeled it as true.

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✔ Corrected Biconditional (Problem 6d):

> Biconditional: If you live in Dallas, then you live in Texas, and if you live in Texas, then you live in Dallas.
> Or: *You live in Dallas if and only if you live in Texas.*
> Truth Value: False
> Reason: While all Dallas residents live in Texas, not all Texas residents live in Dallas (e.g., Houston, Austin). So the reverse is not true.

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📝 Final Grade Suggestion:

Almost perfect! Minor error in biconditional.
Grade: A– or A depending on grading policy.

Let me know if you'd like a cleaned-up version of the worksheet with corrections!