To create a truth table for the expression \((p \rightarrow q) \vee (q \rightarrow r)\), we need to consider the truth values for all possible combinations of the variables \(p\), \(q\), and \(r\).

### Step-by-Step Breakdown:

1. **Implication (\(\rightarrow\))**: The statement \(A \rightarrow B\) (if \(A\), then \(B\)) is false only when \(A\) is true and \(B\) is false. In all other cases, it is true.
2. **Disjunction (\(\vee\))**: The statement \(A \vee B\) (A or B) is true if at least one of \(A\) or \(B\) is true.

Let's break down the components of \((p \rightarrow q) \vee (q \rightarrow r)\) into a truth table:

| \(p\) | \(q\) | \(r\) | \(p \rightarrow q\) | \(q \rightarrow r\) | \((p \rightarrow q) \vee (q \rightarrow r)\) |
|:----:|:----:|:----:|:------------------:|:------------------:|:--------------------------------------------:|
|  T   |  T   |  T   |         T          |         T          |                     T                        |
|  T   |  T   |  F   |         T          |         F          |                     T                        |
|  T   |  F   |  T   |         F          |         T          |                     T                        |
|  T   |  F   |  F   |         F          |         T          |                     T                        |
|  F   |  T   |  T   |         T          |         T          |                     T                        |
|  F   |  T   |  F   |         T          |         F          |                     T                        |
|  F   |  F   |  T   |         T          |         T          |                     T                        |
|  F   |  F   |  F   |         T          |         T          |                     T                        |

### Explanation:
1. **Columns for \(p\), \(q\), and \(r\)**: These list all possible truth values (True, False) for \(p\), \(q\), and \(r\).
2. **Column \(p \rightarrow q\)**: True except when \(p\) is True and \(q\) is False.
3. **Column \(q \rightarrow r\)**: True except when \(q\) is True and \(r\) is False.
4. **Final Column \((p \rightarrow q) \vee (q \rightarrow r)\)**: True when at least one of \(p \rightarrow q\) or \(q \rightarrow r\) is true.

Thus, the compound statement \((p \rightarrow q) \vee (q \rightarrow r)\) is always true for all possible values of \(p\), \(q\), and \(r\).

Would you like more details on any of these steps, or have other questions?

**Related Questions:**
1. What are the truth tables for \(p \rightarrow q\) and \(q \rightarrow r\) individually?
2. How do truth tables relate to logical equivalences?
3. Can you simplify the expression \((p \rightarrow q) \vee (q \rightarrow r)\)?
4. What is the significance of tautologies in logical statements?
5. How do you represent the logical disjunction (\(\vee\)) in different programming languages?

**Tip:** Understanding the properties of logical connectives like implication and disjunction can help in reducing complex logical statements to simpler forms.

Explore the truth table for the logical expression (p→q)∨(q→r) with a detailed breakdown. Understand the roles of implication and disjunction in logic. Suitable for Grades 9-12 students learning logical reasoning.

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