付録 F — How to Write Proofs

F.1 Types of things to Prove

• x = y
• p \implies q
• p \iff q
• \forall x such that [condition], [statement]
• \exists x such that [statement]
• \neg p

F.2 Proof Techniques

• Direct proof: Assume p is true and show that q is true.
• Proof by contradiction: For p \implies q, assume p is true and q is false, and show that this leads to a contradiction.
• Proof by contrapositive: For p \implies q, prove \neg q \implies \neg p instead.
• Proof by induction: For statements involving natural numbers, prove the base case and then show that if the statement holds for n, it also holds for n+1, which is called the inductive step.
• Proof by cases: Break the statement into several cases and prove each case separately.

F.3 Resources

• L. V. Snyder and Z.-J. M. Shen, Fundamentals of supply chain theory, 2nd ed. Nashville, TN: John Wiley & Sons, 2025.
• How to Prove It: A Structured Approach by Daniel J. Velleman, website
• Mathematical Reasoning Writing and Proof, Version 3 by Ted Sundstrom, website