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It looks like you are working on a truth table for the logical statement \(\sim (p \leftrightarrow q)\), where \(\leftrightarrow\) represents the biconditional (if and only if). Let's walk through the steps to complete this truth table.

The logical expression \(\sim (p \leftrightarrow q)\) can be broken down as follows:

1. **Biconditional \(p \leftrightarrow q\)**: This is true when both \(p\) and \(q\) have the same truth values (either both true or both false). Otherwise, it is false.
2. **Negation \(\sim (p \leftrightarrow q)\)**: After calculating \(p \leftrightarrow q\), you negate the result. If \(p \leftrightarrow q\) is true, the negation is false, and if it's false, the negation is true.

Now, let's fill in the truth table based on the standard input values for \(p\) and \(q\):

| \(p\) | \(q\) | \(p \leftrightarrow q\) | \(\sim (p \leftrightarrow q)\) |
|:---:|:---:|:-------------------:|:--------------------------:|
|  T  |  T  |          T          |              F             |
|  T  |  F  |          F          |              T             |
|  F  |  T  |          F          |              T             |
|  F  |  F  |          T          |              F             |

- For \(p = T\) and \(q = T\), \(p \leftrightarrow q\) is true, so \(\sim (p \leftrightarrow q)\) is false.
- For \(p = T\) and \(q = F\), \(p \leftrightarrow q\) is false, so \(\sim (p \leftrightarrow q)\) is true.
- For \(p = F\) and \(q = T\), \(p \leftrightarrow q\) is false, so \(\sim (p \leftrightarrow q)\) is true.
- For \(p = F\) and \(q = F\), \(p \leftrightarrow q\) is true, so \(\sim (p \leftrightarrow q)\) is false.

Make sure to fill in these values on your homework page, and this should complete the table correctly.

Would you like any further details or have any other questions?

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Here are 5 related questions for you:
1. What is the difference between the biconditional (\(\leftrightarrow\)) and conditional (\(\rightarrow\)) operators?
2. How does negation (\(\sim\)) affect the truth value of a statement?
3. Can you construct a truth table for \(p \rightarrow q\)?
4. What is the logical equivalence of \(p \leftrightarrow q\) in terms of conjunction (\(\land\)) and disjunction (\(\lor\))?
5. How would the truth table change if the expression was \(\sim(p \lor q)\) instead?

**Tip**: When creating truth tables, always start by computing the inner expressions and gradually build toward the final result.

Complete the truth table for ∼(p ↔ q)

Learn how to complete a truth table for the logical expression ∼(p ↔ q). Understand biconditional logic and negation with a step-by-step explanation, suitable for students in Grades 9-12.

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