https://file.aiquickdraw.com/mathbot/file/1727995390936lbkmvpqr.jpg

The compound statement you are working with is \((\neg r \land q) \vee \neg p\), and the truth table needs to be completed. Let's break it down step by step.

The columns you need for the truth table are:

1. **p**, **q**, and **r**: Truth values for the individual variables.
2. **\(\neg r\)**: The negation of **r**.
3. **\(\neg r \land q\)**: The conjunction (AND) between \(\neg r\) and **q**.
4. **\(\neg p\)**: The negation of **p**.
5. **(\(\neg r \land q\)) \vee \neg p**: The disjunction (OR) of the previous steps.

### Step-by-step Truth Table:

| p   | q   | r   | \(\neg r\) | \(\neg r \land q\) | \(\neg p\) | \((\neg r \land q) \vee \neg p\) |
| --- | --- | --- | ---------- | ------------------ | ---------- | --------------------------------- |
| T   | T   | T   | F          | F                  | F          | F                                 |
| T   | T   | F   | T          | T                  | F          | T                                 |
| T   | F   | T   | F          | F                  | F          | F                                 |
| T   | F   | F   | T          | F                  | F          | F                                 |
| F   | T   | T   | F          | F                  | T          | T                                 |
| F   | T   | F   | T          | T                  | T          | T                                 |
| F   | F   | T   | F          | F                  | T          | T                                 |
| F   | F   | F   | T          | F                  | T          | T                                 |

Explanation of each step:

1. **\(\neg r\)** is simply the negation of the values in the **r** column.
2. **\(\neg r \land q\)** checks if both \(\neg r\) and **q** are true.
3. **\(\neg p\)** is the negation of **p**.
4. The final column **\((\neg r \land q) \vee \neg p\)** uses the OR operation between \(\neg r \land q\) and \(\neg p\), resulting in true if either or both are true.

This should fill out the table in your image. Let me know if you'd like further clarifications.

Would you like more details on how I broke down each column, or do you have any questions? Here are some related questions to deepen your understanding:

1. How does negation work in a truth table?
2. What is the difference between conjunction (AND) and disjunction (OR)?
3. How does the truth value of a compound statement change when using negation?
4. How can you verify the result of a complex logical statement?
5. Can a truth table ever have all "True" or all "False" results for a compound statement?

**Tip**: When constructing truth tables, work on negations and conjunctions first before applying the OR operator.

Construct a truth table for the given compound statement (¬r∧q)∨¬p.

Learn how to construct a truth table for the compound statement (¬r∧q)∨¬p with detailed steps. Understand the use of negation, conjunction, and disjunction in truth table logic, suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation