https://file.aiquickdraw.com/mathbot/file/1727385281463htbpn4n1.jpg

The statement in the image asks whether the logical expression \((p \land q) \rightarrow p\) is a tautology. 

To analyze this:

- A **tautology** is a logical statement that is always true, regardless of the truth values of its variables.

The given expression is \((p \land q) \rightarrow p\), meaning "if both \(p\) and \(q\) are true, then \(p\) is true."

Let's break it down:
1. **When both \(p\) and \(q\) are true**: The antecedent (\(p \land q\)) is true, and \(p\) is also true. Thus, the implication is true.
2. **When \(p\) is true and \(q\) is false**: The antecedent (\(p \land q\)) becomes false, so the implication is vacuously true (since an implication with a false antecedent is always true).
3. **When \(p\) is false and \(q\) is true**: The antecedent (\(p \land q\)) becomes false, so the implication is again vacuously true.
4. **When both \(p\) and \(q\) are false**: The antecedent (\(p \land q\)) becomes false, and again, the implication is vacuously true.

In all cases, the expression evaluates to **true**, so it **is a tautology**.

### Answer: **True**

Do you have any questions or would you like further details?

Here are 5 related questions to expand on this concept:
1. What is the difference between a tautology and a contradiction in logic?
2. Can you explain what it means for a logical expression to be satisfiable?
3. How does the truth table for \((p \land q) \rightarrow p\) look like?
4. What is the converse of the expression \((p \land q) \rightarrow p\)?
5. How can De Morgan’s laws be applied to simplify logical expressions?

**Tip:** When analyzing logical implications, always check cases where the antecedent is false, as the implication is vacuously true in those situations.

The logical expression (p ∧ q) → p is a tautology. True or False?

Determine whether the logical expression (p ∧ q) → p is a tautology. This problem covers key concepts in propositional logic, including truth tables, tautologies, and implications. Suitable for college-level discrete mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation