https://file.aiquickdraw.com/mathbot/file/17272679006234gu76dyh.jpg

Let's break down the two compound propositions in the image and construct their truth tables.

### (a) \((p \rightarrow q) \leftrightarrow (\neg q \rightarrow \neg p)\)

- \(p \rightarrow q\) is a conditional statement, which is true except when \(p\) is true and \(q\) is false.
- \(\neg q \rightarrow \neg p\) is the contrapositive of \(p \rightarrow q\), which is logically equivalent to \(p \rightarrow q\). So, we expect the entire statement to always be true, but we will verify this with a truth table.

### (b) \((p \oplus q) \land (p \oplus \neg q)\)

- \(p \oplus q\) is the XOR (exclusive or) operation, which is true when exactly one of \(p\) or \(q\) is true.
- \(p \oplus \neg q\) is true when exactly one of \(p\) or \(\neg q\) is true.
- The conjunction \(\land\) of these two expressions will be true only when both are true.

Now, I will construct the truth tables for both (a) and (b).

Let me calculate them.Here are the truth tables for the given compound propositions:

### (a) \((p \rightarrow q) \leftrightarrow (\neg q \rightarrow \neg p)\)

| \(p\)   | \(q\)   | \(p \rightarrow q\) | \(\neg q \rightarrow \neg p\) | \((p \rightarrow q) \leftrightarrow (\neg q \rightarrow \neg p)\) |
|--------|--------|----------------------|------------------------------|------------------------------------------------------------------|
| True   | True   | True                 | False                        | False                                                            |
| True   | False  | False                | True                         | False                                                            |
| False  | True   | True                 | True                         | True                                                             |
| False  | False  | True                 | True                         | True                                                             |

### (b) \((p \oplus q) \land (p \oplus \neg q)\)

| \(p\)   | \(q\)   | \(p \oplus q\) | \(p \oplus \neg q\) | \((p \oplus q) \land (p \oplus \neg q)\) |
|--------|--------|----------------|--------------------|-----------------------------------------|
| True   | True   | False          | True               | False                                   |
| True   | False  | True           | False              | False                                   |
| False  | True   | True           | False              | False                                   |
| False  | False  | False          | True               | False                                   |

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Would you like more details on any part of these solutions?

### 5 Related Questions:
1. How do logical equivalences work in truth tables?
2. What is the difference between XOR (\(\oplus\)) and OR (\(\lor\))?
3. Why is the contrapositive always logically equivalent to the original conditional?
4. Can you explain how conjunction (\(\land\)) affects the truth value of a compound proposition?
5. How do we interpret truth table results in logic circuits?

**Tip**: Always simplify logical expressions using known equivalences to verify complex propositions.

Construct a complete truth table for each of the following compound propositions: (a) (p → q) ↔ (¬q → ¬p) and (b) (p ⊕ q) ∧ (p ⊕ ¬q).

Learn how to construct truth tables for compound propositions involving conditionals and XOR operations. This example covers logical equivalences, contrapositive statements, and conjunctions, suitable for high school students studying logic.

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