Fill in the truth table for proposition q <-> p
p  q
T  T
T  F
F  T
F  F

To fill in the truth table for the proposition \( q \leftrightarrow p \) (which means "q if and only if p"), we evaluate the biconditional statement. The statement \( q \leftrightarrow p \) is true when both \( p \) and \( q \) have the same truth value, either both true or both false.

Here is the filled truth table:

\[
\begin{array}{|c|c|c|}
\hline
p & q & q \leftrightarrow p \\
\hline
T & T & T \\
T & F & F \\
F & T & F \\
F & F & T \\
\hline
\end{array}
\]

Would you like details on how this table is derived or have any other questions?

Here are 5 related questions for further practice:
1. What is the truth table for \( p \leftrightarrow \neg q \)?
2. How does \( p \rightarrow q \) differ from \( p \leftrightarrow q \)?
3. Can you create a truth table for \( \neg(p \leftrightarrow q) \)?
4. What is the truth table for \( (p \vee q) \leftrightarrow (p \wedge q) \)?
5. How would the truth table change for \( p \leftrightarrow (q \vee \neg r) \)?

**Tip:** In logical biconditional statements (\( \leftrightarrow \)), the result is true only when both components share the same truth value.

Explore how to fill in the truth table for the proposition q <-> p (q if and only if p) with a step-by-step explanation. This problem involves understanding propositional logic, biconditional statements, and truth table construction. Suitable for advanced high school students and anyone interested in logical reasoning.

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