Students complete a worksheet on conditional statements, identifying truth values and classifying statements as conditional, converse, contrapositive, or bi-conditional.

It looks like you've uploaded a worksheet involving logic and conditional statements. The text is somewhat blurry, but I can help interpret and solve the problems based on what’s visible.

Let me break it down into two main sections:

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Section 1: Conditional Statements and Truth Values

This section involves understanding logical implications related to rational and whole numbers.

Given Statement:
> If a number is a rational number, then it is also a whole number.

We are asked to analyze:
- Converse
- Inverse
- Contrapositive
- Biconditional

Let’s define:
- $ p $: A number is a rational number.
- $ q $: A number is a whole number.

Original statement: $ p \rightarrow q $

Now let's analyze each:

#### 1. Converse: $ q \rightarrow p $
> If a number is a whole number, then it is a rational number.

- Truth Value: True
- All whole numbers (0, 1, 2, ...) are rational because they can be written as fractions: e.g., $ 2 = \frac{2}{1} $.

#### 2. Inverse: $ \neg p \rightarrow \neg q $
> If a number is not a rational number, then it is not a whole number.

- Truth Value: True
- This is logically equivalent to the contrapositive of the original (but note: the original is false, so we must check carefully).
- However, since the original statement is false, we need to assess this.
- But actually, the inverse is logically equivalent to the converse.
- Since the converse is true, and the inverse has the same truth value as the converse in classical logic? Wait — no: inverse is NOT equivalent to converse.
- Let’s think: If a number is not rational → not whole.
- Suppose a number is irrational (e.g., √2). Then it is not a whole number → True.
- So: if not rational → not whole → True.
- So the inverse is true.

But wait — the original statement "If rational → whole" is false, because there are rational numbers that are not whole numbers (e.g., $ \frac{1}{2} $).

So let’s correct:

- Original: $ p \rightarrow q $: False (because $ \frac{1}{2} $ is rational but not whole)
- Converse: $ q \rightarrow p $: True (whole ⇒ rational)
- Inverse: $ \neg p \rightarrow \neg q $: True (not rational ⇒ not whole) — yes, because all irrationals are not whole numbers.
- Contrapositive: $ \neg q \rightarrow \neg p $: False — because not whole ⇒ not rational?
- Example: $ \frac{1}{2} $ is not whole, but it is rational → So $ \neg q $ is true, $ \neg p $ is false → implication fails → False
- Biconditional: $ p \leftrightarrow q $: False — because the two directions don’t match.

But looking at your image, the truth values seem to have been filled in. Let’s see what you wrote:

You have:
- Converse: True
- Inverse: True
- Contrapositive: True ← This is likely incorrect
- Biconditional: True ← Also incorrect

Wait — the contrapositive of $ p \rightarrow q $ is $ \neg q \rightarrow \neg p $

- $ \neg q $: Not a whole number
- $ \neg p $: Not a rational number

So: If a number is not a whole number → then it is not rational?

But $ \frac{1}{2} $ is not a whole number, yet it is rational → So the implication fails → False

So contrapositive is False, not True.

And biconditional: $ p \leftrightarrow q $: only true if both directions are true → but original is false, converse is true → so overall False

So corrections:
- Original: False
- Converse: True
- Inverse: True
- Contrapositive: False
- Biconditional: False

But perhaps the original was meant to be "If a number is a whole number, then it is rational" — which would make more sense.

But as written: “If a number is rational, then it is whole” — this is false.

So the truth values should be:
- Converse: True
- Inverse: True
- Contrapositive: False
- Biconditional: False

But your image says:
- Contrapositive: True ← Incorrect
- Biconditional: True ← Incorrect

So likely a mistake in the answers.

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Section 2: Logical Statements with p and q

Let:
- $ p $: You have a library card
- $ q $: You can check out books

We are given several logical forms and need to write the English sentence and classify it.

#### 4. $ p \rightarrow q $
> If you have a library card, then you can check out books.

- Classify: Conditional

✔ Correct.

#### 5. $ \neg q \rightarrow \neg p $
> If you cannot check out books, then you do not have a library card.

- This is the contrapositive of $ p \rightarrow q $
- Contrapositive is logically equivalent to the original.
- So: Contrapositive

✔ Correct.

#### 6. $ q \rightarrow p $
> If you can check out books, then you have a library card.

- This is the converse of $ p \rightarrow q $
- Converse

✔ Correct.

#### 7. $ \neg p \rightarrow \neg q $
> If you don't have a library card, then you cannot check out books.

- This is the inverse of $ p \rightarrow q $
- Inverse

✔ Correct.

#### 8. $ p \leftrightarrow q $
> If you have a library card, then you can check out books, and if you can check out books, then you have a library card.

- This is the biconditional
- Biconditional

✔ Correct.

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✔ Final Answers Summary

#### Section 1: Rational & Whole Numbers

Let’s assume the original statement is:
> If a number is rational, then it is a whole number. → False

Then:

| Form | Statement | Truth Value |
|------------------|---------------------------------------------------------------------------|-------------|
| Original | If rational → whole | False |
| Converse | If whole → rational | True |
| Inverse | If not rational → not whole | True |
| Contrapositive | If not whole → not rational | False |
| Biconditional | Rational ↔ whole | False |

> ⚠️ Note: Your handwritten answers say "True" for contrapositive and biconditional — these are incorrect.

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#### Section 2: Library Card Logic

| Symbolic | Statement | Type |
|---------|---------------------------------------------------------------------------|----------------|
| 4. $ p \rightarrow q $ | If you have a library card, then you can check out books. | Conditional |
| 5. $ \neg q \rightarrow \neg p $ | If you cannot check out books, then you do not have a library card. | Contrapositive |
| 6. $ q \rightarrow p $ | If you can check out books, then you have a library card. | Converse |
| 7. $ \neg p \rightarrow \neg q $ | If you don't have a library card, then you cannot check out books. | Inverse |
| 8. $ p \leftrightarrow q $ | If you have a library card, then you can check out books, and vice versa. | Biconditional |

✔ All classifications are correct.

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🔍 Final Notes

- The truth values in Section 1 need correction.
- The logical forms in Section 2 are correctly identified.

Would you like me to provide a corrected version of the worksheet or explain any part in more detail?