Construct truth tables for the following statements:
 i.(p →~p) ∨(p→q):


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

**(p → ¬p) ∨ (p → q)**

### Steps:
1. Break the statement into components:
   - \( p \)
   - \( \neg p \) (negation of \( p \))
   - \( p \rightarrow \neg p \) (implication)
   - \( p \rightarrow q \) (implication)
   - \( (p \rightarrow \neg p) \lor (p \rightarrow q) \) (disjunction)

### Truth table:

| \( p \) | \( q \) | \( \neg p \) | \( p \rightarrow \neg p \) | \( p \rightarrow q \) | \( (p \rightarrow \neg p) \lor (p \rightarrow q) \) |
|--------|--------|--------------|----------------------------|-----------------------|-----------------------------------------------------|
|   T    |   T    |      F       |             F              |          T            |                          T                          |
|   T    |   F    |      F       |             F              |          F            |                          F                          |
|   F    |   T    |      T       |             T              |          T            |                          T                          |
|   F    |   F    |      T       |             T              |          T            |                          T                          |

### Explanation:
1. \( p \rightarrow \neg p \): This is true only when \( p \) is false, as a true proposition cannot imply its own negation.
2. \( p \rightarrow q \): This is true when either \( p \) is false, or when both \( p \) and \( q \) are true.
3. \( (p \rightarrow \neg p) \lor (p \rightarrow q) \): The disjunction of these two implications is true whenever at least one of the implications is true.

### Result:
The truth table shows that the overall statement \( (p \rightarrow \neg p) \lor (p \rightarrow q) \) is only false when \( p = T \) and \( q = F \).

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Do you want further clarification or details on any step?

Here are five related questions:
1. What is the truth table for \( \neg(p \lor q) \)?
2. How does the truth table change if we swap \( \neg p \) with \( \neg q \)?
3. How do we construct a truth table for three variables, like \( p \), \( q \), and \( r \)?
4. Can you construct a truth table for \( p \rightarrow (q \rightarrow p) \)?
5. What is the difference between implication and bi-conditional statements?

**Tip:** In logical implications, \( p \rightarrow q \) is always true if \( p \) is false, regardless of \( q \).

Construct truth tables for the following statements: i. (p → ~p) ∨ (p→q)

Learn how to construct a truth table for the logical statement (p → ¬p) ∨ (p → q), exploring the key concepts of logical implication, negation, and disjunction. Detailed steps and explanations are provided, suitable for students in Grades 9-12.

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