Construct a truth table for the following symbolic statement.
left parenthesis p right arrow q right parenthesis left right arrow left parenthesis q right arrow tilde p right parenthesis(p → q) ↔ (q → ~p)
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Part 1
Use the type of truth table where each variable or operator in the given expression has its own column. Fill in the rows of the truth table one at a time. 
p
q
​(pp
right arrow→
qq​)
left right arrow↔
​(qq
right arrow→
tilde p~p​)
T
T
▼ 
T
F
▼ 
F
T
▼ 
T
F
▼ 
F
T
▼ 
T
F
▼ 
F
T
▼ 
T
F


Let's construct the truth table for the symbolic statement:

\[
(p \to q) \leftrightarrow (q \to \neg p)
\]

We'll follow the steps systematically, creating a column for each variable and operator. The columns will be as follows:

1. \( p \) (the first variable)
2. \( q \) (the second variable)
3. \( p \to q \) (implication of \( p \) and \( q \))
4. \( \neg p \) (negation of \( p \))
5. \( q \to \neg p \) (implication of \( q \) and \( \neg p \))
6. \( (p \to q) \leftrightarrow (q \to \neg p) \) (biconditional between the two implications)

Now, we'll fill in the truth table with the possible truth values for \( p \) and \( q \), and then calculate each column.

### Truth Table

| \( p \) | \( q \) | \( p \to q \) | \( \neg p \) | \( q \to \neg p \) | \( (p \to q) \leftrightarrow (q \to \neg p) \) |
|--------|--------|---------------|------------|-------------------|--------------------------------------------|
|   T    |   T    |       T       |     F      |         F         |                    T                       |
|   T    |   F    |       F       |     F      |         T         |                    F                       |
|   F    |   T    |       T       |     T      |         T         |                    T                       |
|   F    |   F    |       T       |     T      |         T         |                    T                       |

### Explanation:

1. **\( p \to q \)** (Implication):
   - If \( p \) is true and \( q \) is true, then \( p \to q \) is true.
   - If \( p \) is true and \( q \) is false, then \( p \to q \) is false.
   - If \( p \) is false, \( p \to q \) is always true (because a false premise implies anything).

2. **\( \neg p \)** (Negation of \( p \)):
   - This is simply the opposite of \( p \). If \( p \) is true, \( \neg p \) is false, and vice versa.

3. **\( q \to \neg p \)** (Implication \( q \) implies \( \neg p \)):
   - Similar to the implication for \( p \to q \), but with \( q \) and \( \neg p \).

4. **\( (p \to q) \leftrightarrow (q \to \neg p) \)** (Biconditional):
   - The biconditional is true when both sides have the same truth value (both true or both false), and false otherwise.

The final column shows the truth values for the entire expression.

---

Would you like more details on any specific part of the table? Here are some follow-up questions to deepen your understanding:

1. How is the truth value of an implication (\( p \to q \)) determined?
2. Why does a false \( p \) always make the implication \( p \to q \) true?
3. Can you identify cases where the biconditional \( \leftrightarrow \) is false?
4. What happens if both \( p \) and \( q \) are false in the expression?
5. How would the truth table change if we used \( p \leftrightarrow q \) instead of the given expression?

**Tip:** When working with truth tables, remember that an implication is only false when the first part is true, and the second part is false.

Construct a truth table for the following symbolic statement: (p → q) ↔ (q → ~p). Use the type of truth table where each variable or operator in the given expression has its own column. Fill in the rows of the truth table one at a time.

Learn how to construct a truth table for the logical expression (p → q) ↔ (q → ~p). This example covers key concepts in propositional logic, including implications, negations, and biconditionals. Suitable for students studying Discrete Mathematics.

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