https://file.aiquickdraw.com/mathbot/file/1726689715337m6gownrj.jpg

The image shows a truth table that needs to be completed for an argument. The table includes two premises and a conclusion, with columns for different truth values of \( A \) and \( B \).

### The structure of the truth table:
- **Premise 1**: \( \neg A \iff B \)
- **Premise 2**: \( A \rightarrow \neg B \)
- **Conclusion**: \( \neg A \)

You need to evaluate these statements for all possible combinations of \( A \) and \( B \) (i.e., when \( A \) and \( B \) take the values 0 or 1).

### Truth Table Guide:
The truth table should be completed by evaluating each row for the premises and the conclusion.

#### Symbols:
- \( \neg \): NOT (negation)
- \( \iff \): Biconditional (if and only if)
- \( \rightarrow \): Implication (if... then...)

### Steps to complete the truth table:

1. **Premise 1: \( \neg A \iff B \)**
   - This is true when both \( \neg A \) and \( B \) are the same (both true or both false).
   
2. **Premise 2: \( A \rightarrow \neg B \)**
   - This is false only if \( A \) is true and \( \neg B \) is false. Otherwise, it's true.

3. **Conclusion: \( \neg A \)**
   - This simply negates \( A \).

The truth values for all combinations of \( A \) and \( B \) will look like:

| A | B | \( \neg A \iff B \) | \( A \rightarrow \neg B \) | \( \neg A \) |
|---|---|--------------------|-------------------------|------------|
| 0 | 0 |                      |                          |            |
| 0 | 1 |                      |                          |            |
| 1 | 0 |                      |                          |            |
| 1 | 1 |                      |                          |            |

### Filling in the table:
1. Evaluate \( \neg A \) for each row.
2. Check \( \neg A \iff B \) (Premise 1).
3. Check \( A \rightarrow \neg B \) (Premise 2).
4. Verify the conclusion \( \neg A \).

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Do you want me to fill in the table fully or explain any part of this in more detail? Feel free to ask for specific clarifications.

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Here are some related questions for further learning:
1. What is the truth value of \( \neg A \iff B \) when \( A = 0 \) and \( B = 1 \)?
2. How do you determine if an argument is valid or invalid using a truth table?
3. What does it mean when a row in a truth table is a counterexample?
4. How does the implication \( A \rightarrow \neg B \) behave when both \( A \) and \( B \) are true?
5. What is the difference between a valid and a sound argument?

**Tip:** In a truth table, an argument is valid if there is no row where all premises are true but the conclusion is false.

Please complete the truth table for the following argument and answer the questions below:
Premise 1: (¬A ≡ B)
Premise 2: (A → ¬B)
Conclusion: (¬A).

Learn how to construct and complete a truth table for a logical argument involving biconditional (≡) and implication (→) with detailed explanations of premises and conclusion. Suitable for high school students exploring propositional logic.

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