1. Statement of the problem. Problem: Decide whether a given propositional formula is a tautology. 2. Definition and rule. A formula $\varphi$ is a tautology if it is true under every truth assignment to its propositional variables. Equivalently, $\varphi$ is a tautology iff $\models \varphi$. 3. Methods to check and important rules. Method A (truth table): list all $2^n$ assignments for $n$ variables and evaluate $\varphi$ on each assignment; if every evaluation is true then $\varphi$ is a tautology. Method B (algebraic/logical equivalences): apply valid equivalences such as De Morgan, distribution, implication rewrites, and simplification to transform $\varphi$ into a known tautology. Method C (SAT/semantic): $\varphi$ is a tautology precisely when $\lnot\varphi$ is unsatisfiable; so run a SAT check on $\lnot\varphi$. 4. Intermediate-work example (law of excluded middle). Take $\varphi = p \lor \lnot p$. If $p$ is true then $p \lor \lnot p$ evaluates to true. If $p$ is false then $\lnot p$ is true, so $p \lor \lnot p$ evaluates to true. Since both possible assignments give true, $p \lor \lnot p$ is a tautology. 5. Counterexample example (contradiction) and check process. Take $\psi = p \land \lnot p$. If $p$ is true then $\lnot p$ is false, so $p \land \lnot p$ is false. If $p$ is false then $p$ is false, so $p \land \lnot p$ is false. Every assignment gives false, so $p \land \lnot p$ is not a tautology (it is a contradiction). 6. Practical algorithm to apply when given a concrete formula. Step 1: Identify propositional variables and pick a method (truth table for small $n$, SAT or equivalences for larger $n$). Step 2: Execute the method and record intermediate evaluations or equivalence steps showing simplification. Step 3: Conclude: if any assignment makes the formula false then it is not a tautology; if none do then it is a tautology. Final answer: You cannot assert "tautology" without a specific formula; some formulas are tautologies (for example $p \lor \lnot p$) and others are not (for example $p \land \lnot p$). If you provide a particular formula I will check it step by step.