Trivial and Vacuous Proofs

• Read pages 77 through 80 (section 3.1)
• Some key questions to answer:
  1. What is the difference between axioms and theorems?
  2. What other terms do we use in place of the word theorem? What two factors determine our choice of word?
  3. What is a corollary, and what is a lemma?
  4. Which quantified statement do we mean when we write: “for \(x\in S\), if \(P(x)\) then \(Q(x)\)”?
  5. When do we say that our proof of a result “for \(x\in S\), if \(P(x)\) then \(Q(x)\)” is a trivial proof?
  6. Give examples of results with trivial proofs.
  7. When do we say that our proof of a result “for \(x\in S\), if \(P(x)\) then \(Q(x)\)” is a vacuous proof?
  8. Give examples of results with vacuous proofs.
  9. Explain the main difference between trivial proofs and vacuous proofs.
  10. When can a result have both a trivial proof and a vacuous proof?
• Practice problems from section 3.1 (page 93). In each of these problems, make sure you identify whether you useda trivial proof or a vacuous proof: 3.1, 3.3, 3.5