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To complete this truth table for the implication \((A \vee B) \rightarrow \neg C\), let's break it down step by step.

The table has the following columns:

1. **A**, **B**, and **C**: These are the given truth values.
2. **A ∨ B**: This is the logical OR of A and B.
3. \((A \vee B) \rightarrow \neg C\): This is the implication where \((A \vee B)\) implies \(\neg C\) (the negation of C).

We'll follow the steps to compute the truth values for the expression \((A \vee B) \rightarrow \neg C\).

### Step-by-step procedure:
1. **A ∨ B**: For each row, perform the OR operation between A and B.
2. **Negation of C** (\(\neg C\)): Compute the negation of C.
3. **Implication** \((A \vee B) \rightarrow \neg C\): Use the implication rule: the implication \(P \rightarrow Q\) is false only when \(P\) is true and \(Q\) is false, otherwise it is true.

Let's fill in the table:

| A   | B   | C   | A ∨ B | ¬C  | (A ∨ B) → ¬C |
|-----|-----|-----|-------|-----|--------------|
| T   | T   | T   |   T   |  F  |      F       |
| T   | T   | F   |   T   |  T  |      T       |
| T   | F   | T   |   T   |  F  |      F       |
| T   | F   | F   |   T   |  T  |      T       |
| F   | T   | T   |   T   |  F  |      F       |
| F   | T   | F   |   T   |  T  |      T       |
| F   | F   | T   |   F   |  F  |      T       |
| F   | F   | F   |   F   |  T  |      T       |

### Explanation of key rows:
- Row 1: \(A = T\), \(B = T\), \(C = T\). Since \(A \vee B = T\) and \(\neg C = F\), the implication is false.
- Row 7: \(A = F\), \(B = F\), \(C = T\). Since \(A \vee B = F\) and \(\neg C = F\), the implication is true (because a false antecedent makes the implication true).

Would you like any further clarification?

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### Related questions:
1. How is the logical OR (A ∨ B) calculated in a truth table?
2. What is the rule for a logical implication \(P \rightarrow Q\)?
3. Can you explain how negation works in logic (\(\neg C\))?
4. How would the truth table change if the expression was \((A \wedge B) \rightarrow \neg C\)?
5. Can you show an example where the implication is always true?

### Tip:
When working with logical implications, remember that the only time the implication is false is when the antecedent (the part before the arrow) is true and the consequent (the part after the arrow) is false.

Complete the truth table for the implication (A ∨ B) → ¬C.

Learn how to complete the truth table for the implication (A ∨ B) → ¬C, including step-by-step solutions involving logical disjunction, negation, and implication. This problem involves propositional logic and is suitable for college-level students.

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