To determine whether the given statement $[ (p \to q) \land (q \to r) ] \to (p \to r)$ is a tautology, let's analyze it step by step. A statement is a tautology if it is true for all possible truth values of its propositional variables.

Step 1: Expressing the statement

We are given: $[ (p \to q) \land (q \to r) ] \to (p \to r)$

Step 2: Breaking down the components

1. $p \to q$ is equivalent to $\neg p \lor q$.
2. $q \to r$ is equivalent to $\neg q \lor r$.
3. $p \to r$ is equivalent to $\neg p \lor r$.

The statement becomes: $[ (\neg p \lor q) \land (\neg q \lor r) ] \to (\neg p \lor r)$

Step 3: Analyzing the logical equivalence

The implication $A \to B$ is equivalent to $\neg A \lor B$. Applying this, the entire expression becomes: $\neg [ (\neg p \lor q) \land (\neg q \lor r) ] \lor (\neg p \lor r)$

Step 4: Truth table analysis

Constructing a truth table for $p$, $q$, and $r$, and evaluating each component systematically:

$p$	$q$	$r$	$\neg p \lor q$	$\neg q \lor r$	$(\neg p \lor q) \land (\neg q \lor r)$	$\neg p \lor r$	$[ (\neg p \lor q) \land (\neg q \lor r) ] \to (\neg p \lor r)$
T	T	T	T	T	T	T	T
T	T	F	T	F	F	F	T
T	F	T	F	T	F	T	T
T	F	F	F	F	F	F	T
F	T	T	T	T	T	T	T
F	T	F	T	F	F	T	T
F	F	T	T	T	T	T	T
F	F	F	T	F	F	T	T

Step 5: Conclusion

From the truth table, the statement evaluates to true for all possible truth values of $p$, $q$, and $r$. Hence, the given statement is a tautology.

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Would you like me to go over any part of the process or elaborate on specific logical steps?

Further Exploration

1. How to systematically evaluate implications in logical statements?
2. What is the role of truth tables in proving tautologies?
3. Could we use logical equivalences to simplify this problem further?
4. What are common examples of tautologies in propositional logic?
5. How does this result apply to real-world logical reasoning?

Tip

When analyzing logical statements, always try simplifying them using equivalences like $\neg (A \land B) \equiv \neg A \lor \neg B$ or $A \to B \equiv \neg A \lor B$ to make the analysis more straightforward.