https://file.aiquickdraw.com/mathbot/file/1727099888559mty2fr65.jpg

The problem in the image is asking to create a truth table for the following compound statement:

**\( p \Rightarrow (p \Rightarrow q) \)**

Let's break this down and create the truth table.

### Components:
1. **\( p \)**: A simple proposition.
2. **\( q \)**: Another simple proposition.
3. **\( p \Rightarrow q \)**: This is an implication statement, which is true unless \( p \) is true and \( q \) is false.
4. **\( p \Rightarrow (p \Rightarrow q) \)**: This is the main compound statement for which we need to create the truth table.

### Truth Table Construction:

We'll need four columns: \( p \), \( q \), \( p \Rightarrow q \), and \( p \Rightarrow (p \Rightarrow q) \).

| \( p \) | \( q \) | \( p \Rightarrow q \) | \( p \Rightarrow (p \Rightarrow q) \) |
|--------|--------|----------------------|-------------------------------------|
|   T    |   T    |          T           |                T                    |
|   T    |   F    |          F           |                F                    |
|   F    |   T    |          T           |                T                    |
|   F    |   F    |          T           |                T                    |

### Explanation:
1. **\( p \Rightarrow q \)**: This column shows the result of the implication \( p \Rightarrow q \).
   - When \( p = T \) and \( q = T \), \( p \Rightarrow q \) is True.
   - When \( p = T \) and \( q = F \), \( p \Rightarrow q \) is False.
   - When \( p = F \), \( p \Rightarrow q \) is always True, regardless of \( q \).
   
2. **\( p \Rightarrow (p \Rightarrow q) \)**: The final column shows the main statement's truth value.
   - When \( p = T \), the statement depends on \( p \Rightarrow q \). If \( p \Rightarrow q \) is true, then \( p \Rightarrow (p \Rightarrow q) \) is true; otherwise, it's false.
   - When \( p = F \), \( p \Rightarrow (p \Rightarrow q) \) is always true, because any implication with a false antecedent is true.

Would you like a more detailed breakdown of the steps? Also, here are five related questions to extend this concept:

1. What is the truth table for \( q \Rightarrow (p \Rightarrow q) \)?
2. How does the truth table of \( p \wedge q \) compare to \( p \Rightarrow q \)?
3. Can you express \( p \Rightarrow (p \Rightarrow q) \) using logical operators like \( \neg \) (NOT) and \( \vee \) (OR)?
4. How does a biconditional \( p \iff q \) differ from \( p \Rightarrow q \)?
5. What is the difference between the logical operations \( \wedge \) (AND) and \( \Rightarrow \) (IMPLICATION)?

**Tip**: When analyzing compound statements in logic, always break them down into smaller sub-statements to simplify the truth table construction process.

Buatlah tabel kebenaran dari pernyataan majemuk berikut: p ⇒ (p ⇒ q)

Create the truth table for the compound statement p ⇒ (p ⇒ q). This problem involves understanding propositional logic, implication, and truth table construction, suitable for students in Grades 10-12.

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