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Proof Strategies

Direct, contrapositive, induction, contradiction, counting, and inequalities.

Mathematical Proof Strategies Handbook

Mathematical Principles & Theorems

Formalizes foundational deductive proof paradigms: (1) Direct Proof: assumes premise \(P\) and applies definitions/axioms to derive \(Q\); (2) Proof by Contraposition: establishes \(\neg Q \implies \neg P\), logically equivalent to \(P \implies Q\); (3) Proof by Contradiction: assumes \(P \land \neg Q\) and derives a logical absurdity \(\bot\) (e.g., Hippasus proof that \(\sqrt{2} \notin \mathbb{Q}\)); (4) Mathematical Induction: proves base case \(P(0)\) and inductive step \(\forall k, P(k) \implies P(k+1)\).

Operating Instructions

  • Select a classic mathematical theorem from the theorem catalog.
  • Choose proof strategy: Direct, Contradiction, Induction, or Contrapositive.
  • Step through sequential deductive derivation cards with highlighted logical connectives.
  • Inspect graphical proof diagrams illustrating geometric or inductive visual representations.

Direct Proof

Assume P is true, then through a logical sequence of steps, show Q must also be true.

Direct Proof