Proving Biconditionals

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• Questions?
• Proving Biconditionals
  • Last part of section 3.2
  • Examples / Problems
    • Formal/prose proofs?
      • See Example from below
    • To prove P ↔ Q prove P → Q and prove Q → P
    • For all integers n, n is even if and only if n/2 is an integer
      • Part 1
        • Assume n even
        • n = 2q, for integer q
        • n/2 = q
      • Part 2 (if n/2 is integer, then n is even)
        • Proof via contrapositive
        • If n odd, then n/2 not integer
        • n = 2r + 1, for integer r
        • n/2 = r + 1/2
        • Integer + 1/2 not integer
      • Part 2 alternate
        • Assume n/2 integer
        • n/2 = k
        • n = 2k
        • This is even by definition of even
      • See this as a nicely written formal proof with LaTeX source
    • For all real numbers x, 2x - 3 > 0 if and only if x > 3/2
      • Part 1 if 2x - 3 > 0 then x > 3/2
        • Algebra: 2x - 3 > 0
        • Implies 2x > 3
        • Implies x > 3/2
      • Part 2 if x > 3/2 then 2x - 3 > 0
        • Assume x > 3/2
        • So 2x > 3
        • So 2x - 3 > 0
      • Or
        • 2x - 3 > 0
        • iff 2x > 3
        • iff x > 3/2
      • Moral: In those cases where each step of a proof relies on its own biconditional, you can do both directions at once
    • For all real numbers x, x = 4 if and only if x³ = 64 if and only if x^3/2 = ±8
      • x = 4
      • Implies x³ = 4³ = 64
      • Implies x^3/2 = √64 = ±8
      • Implies √x = ±2 so x = 4
      • Idea
        • Prove P ↔ Q ↔ R ↔ … ↔ Z
        • By proving cycle of implications P → Q → R → … → Z → P
• Next
  • Proof by contradiction
  • Section 3.3

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