The Art of the Proof: Core Methods [2/7]

# Episode 2: The Core Methods**Objective:** To master the three canonical methods of proof—Direct, Contrapositive, and Contradiction—and understand the strategic logic for choosing between them [Part II, 4].### I. Introduction: The Architect’s Blueprint### II. Segment 1: The Direct March (Direct Proof)* **Defining the Direct Proof:** * **The Forward-Backward Method:*** **Forward from Hypothesis:** * **Backward from Conclusion:** * **Bridging the Gap:** * **Proving Universal Conditionals:** * **The "Witness" in Existence:** * **Common Direct Pitfalls:** ### III. Segment 2: Turning the Problem Around (Proof by Contrapositive)* **The Logic of Equivalence:** * **When to Flip:** * **The Classic Example:** * **Contrapositive vs. Contradiction:** Learning to recognize when a "contradiction" proof is actually just a "contrapositive wearing a heavy coat"### IV. Segment 3: The Impossible Opposite (Proof by Contradiction)* **The Collapse of the Negation:** * **Handling Conditionals:** * **Classic Case Study 1: The Irrationality of The Square Root of 2:*** **Classic Case Study 2: Euclid’s Infinitude of Primes:*** **The Danger of the Wrong Negation:** ### V. Conclusion: Choosing Your Weapon* **The Decision Heuristic:*** Try **Direct** first for positive, unpackable statements.* Use **Contrapositive** for negative conclusions or messy algebra.* Reserve **Contradiction** for non-existence claims or when the other two fail.* **Preview of Episode 3:** Refining the toolkit with Case Analysis, "Without Loss of Generality" (WLOG), and the power of Counterexamples.