https://file.aiquickdraw.com/mathbot/file/1723666934472db6cqg3g.jpg

Let's break down the logical expression provided in the image, which is:

\[ q \leftrightarrow (\lnot p \lor \lnot q) \]

This is a biconditional statement, which is true when both sides have the same truth value. To analyze it, we’ll use a truth table to evaluate all possible combinations of truth values for \( p \) and \( q \).

### Truth Table:

| \( p \) | \( q \) | \( \lnot p \) | \( \lnot q \) | \( \lnot p \lor \lnot q \) | \( q \leftrightarrow (\lnot p \lor \lnot q) \) |
|---------|---------|---------------|---------------|----------------------------|--------------------------------------------|
|    T    |    T    |       F       |       F       |              F             |                      F                     |
|    T    |    F    |       F       |       T       |              T             |                      T                     |
|    F    |    T    |       T       |       F       |              T             |                      F                     |
|    F    |    F    |       T       |       T       |              T             |                      T                     |

### Explanation:

1. **When \( p = T \) and \( q = T \):**
   - \( \lnot p = F \), \( \lnot q = F \)
   - \( \lnot p \lor \lnot q = F \)
   - \( q \leftrightarrow (\lnot p \lor \lnot q) = F \leftrightarrow F = T \) (Incorrect: should be False)

2. **When \( p = T \) and \( q = F \):**
   - \( \lnot p = F \), \( \lnot q = T \)
   - \( \lnot p \lor \lnot q = T \)
   - \( q \leftrightarrow (\lnot p \lor \lnot q) = F \leftrightarrow T = F \)

3. **When \( p = F \) and \( q = T \):**
   - \( \lnot p = T \), \( \lnot q = F \)
   - \( \lnot p \lor \lnot q = T \)
   - \( q \leftrightarrow (\lnot p \lor \lnot q) = T \leftrightarrow T = T \)

4. **When \( p = F \) and \( q = F \):**
   - \( \lnot p = T \), \( \lnot q = T \)
   - \( \lnot p \lor \lnot q = T \)
   - \( q \leftrightarrow (\lnot p \lor \lnot q) = F \leftrightarrow T = F \)

From the truth table, we see that the expression \( q \leftrightarrow (\lnot p \lor \lnot q) \) is not a tautology or contradiction; it's true for some inputs and false for others.

**Do you want more details or have any questions? Here are some related questions to explore:**

1. What is the difference between a tautology and a contradiction in logic?
2. How do logical operators affect the outcome of truth tables?
3. Can you explain De Morgan's laws in logic?
4. How does the biconditional operator differ from the conditional operator?
5. What are some applications of truth tables in computer science?
6. How do logical expressions relate to boolean algebra?
7. What role do logical connectives play in programming?
8. Can you provide examples of logical equivalences in mathematics?

**Tip:** Familiarizing yourself with truth tables can simplify the understanding and validation of logical statements.

Learn how to analyze and resolve the biconditional logical statement q ↔ (¬p ∨ ¬q) using truth tables and logical operators. This problem explores key concepts in logic, including truth values and biconditional statements. Suitable for advanced high school students and anyone interested in logical reasoning.

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