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.