Within 5 minutes, construct the truth table of [~r V (r→~q)] (~q→~r) Check ALL the options below after constructing the truth table.

I AM DONE CONSTRUCTING THE TRUTH TABLE.

I FINISHED MAKING THE TRUTH TAble

Let's break down the given logical expression: 

**[~r ∨ (r → ~q)] ∧ (~q → ~r)**

To construct the truth table, we'll need to compute the following step by step:

1. Negation of \(r\), denoted as **~r**.
2. Implication \(r → ~q\).
3. Disjunction of **~r** and **(r → ~q)**.
4. Implication \(~q → ~r\).
5. The conjunction of **[~r ∨ (r → ~q)]** and **(~q → ~r)**.

We'll use three variables: \(r\), \(q\), and their negations, leading to 8 rows in the truth table (since \(2^3 = 8\)).

### Step-by-Step Table Construction:

| \(r\) | \(q\) | \(~r\) | \(~q\) | \(r → ~q\) | \(~r ∨ (r → ~q)\) | \(~q → ~r\) | \([~r ∨ (r → ~q)] ∧ (~q → ~r)\) |
|:-----:|:-----:|:------:|:------:|:----------:|:-----------------:|:-----------:|:---------------------------------:|
|   T   |   T   |   F    |   F    |     F      |         F         |      T      |               F                 |
|   T   |   F   |   F    |   T    |     T      |         T         |      F      |               F                 |
|   F   |   T   |   T    |   F    |     T      |         T         |      T      |               T                 |
|   F   |   F   |   T    |   T    |     T      |         T         |      T      |               T                 |
|   T   |   T   |   F    |   F    |     F      |         F         |      T      |               F                 |
|   T   |   F   |   F    |   T    |     T      |         T         |      F      |               F                 |
|   F   |   T   |   T    |   F    |     T      |         T         |      T      |               T                 |
|   F   |   F   |   T    |   T    |     T      |         T         |      T      |               T                 |

### Final Truth Table Explanation:

- **~r** is the negation of \(r\).
- **~q** is the negation of \(q\).
- \(r → ~q\) represents the implication that if \(r\) is true, then \(q\) must be false (i.e., \(~q\) must be true).
- **~r ∨ (r → ~q)** is the disjunction (OR) of ~r and \(r → ~q\).
- **~q → ~r** is the implication that if \(~q\) is true, then \(~r\) must also be true.
- **[~r ∨ (r → ~q)] ∧ (~q → ~r)** is the final conjunction (AND) of both parts.

Now, based on this truth table, you can evaluate any options or check your answers.

Do you have any follow-up questions, or would you like further details?

### Related questions:
1. What is the definition of disjunction in logic?
2. How is the implication rule applied in a truth table?
3. How does negation affect a logical statement's truth value?
4. How do we interpret a conjunction in propositional logic?
5. What is the significance of truth tables in logic?

### Tip: 
Always check the truth values of both components of a conjunction; both must be true for the overall expression to be true.

Within 5 minutes, construct the truth table of [~r V (r→~q)] (~q→~r) Check ALL the options below after constructing the truth table.

Learn how to construct the truth table for the propositional logic expression [~r ∨ (r → ~q)] ∧ (~q → ~r). This detailed breakdown covers key concepts such as negation, implication, disjunction, and conjunction, following the basic rules of propositional logic. Suitable for students in Grades 10-12.

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