The expression \[(\sim r \land \sim s) \rightarrow (\sim r \land \sim s)\] is a **tautology**, which means it is always true. Here's how:

1. **Analyze the left-hand side (premise):**  
   The expression \(\sim r \land \sim s\) means "not \(r\) AND not \(s\)." This will be true only when both \(r\) is false and \(s\) is false.

2. **Analyze the right-hand side (conclusion):**  
   The expression \(\sim r \land \sim s\) is the same as the premise. This means that the conclusion is identical to the premise.

3. **Implication Truth Table:**
   The implication \(A \rightarrow B\) is only false when \(A\) is true and \(B\) is false. Since the premise and conclusion are identical, the implication will always be true because whenever the premise holds, the conclusion will also hold.

Thus, the statement \((\sim r \land \sim s) \rightarrow (\sim r \land \sim s)\) is always true in every case where both \(r\) and \(s\) are false, and the implication still holds for other truth values due to the nature of logical implication. Therefore, it is a tautology.

Would you like a step-by-step truth table for verification or more details?

---

Here are 5 related questions:

1. How do truth tables help in determining if a logical statement is a tautology?
2. What is the difference between a tautology and a contradiction?
3. How would the result change if we swapped the implication with a biconditional?
4. Can you provide examples of other logical tautologies?
5. How can De Morgan’s laws be applied to simplify logical expressions like this?

**Tip:** In logic, the truth of an implication is guaranteed if the premise is false, regardless of the conclusion.

Explore the logical tautology of the expression (~r ∧ ~s) → (~r ∧ ~s). This problem covers key concepts in propositional logic, including the truth table analysis of implication and logical AND. Suitable for students in 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