Discrete Structures

Lecture Notes, 8 Oct 2010

Last Update: 8 October 2010

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§§1.6: Proof Strategies

  1. Top-Level Proof Strategy:

    1. Look at the logical structure
        (i.e., at the principal connective or quantifier)
      of the proposition to be proved:

      • Is it ∀, ∃, →, ¬, etc.?

    2. Then use a lower-level strategy for proving propositions of that form.

  2. A General Strategy Applicable at Any Time:

  3. Direct Proof:

  4. To show ∀x[P(x) → Q(x)]:

  5. An Example:

    1. Prove: If n is odd, then n+1 is even.
      Hidden assumption: n is an integer
      Logical form (FOL transation):
        (*)   (∀ integer n)[Odd(n) → Even(n+1)]

      Strategy & Proof:

      • Proofs are stories;
        you don't just want to say what happens,
        but also how & why

        Let c be an arbitrary integer.

        • i.e., instead of trying to show that every odd integer is such that its successor is even,
          just show it for one "typical" or "arbitrary" integer.

        Show Odd(c) → Even(c+1) (and then use UG to show (*))

          Suppose Odd(c) & show Even(c+1):

          • i.e., instead of showing that the successor of odd c is even,
            suppose an extra fact, namely, that c is odd,
            & just show that c+1 is even

            i.e., show ∃j[c+1=2j] (replace "Even" by its definition)
            But Odd(c)
            ∴ ∃k[c=2k+1] (by def of Odd)

            • (using "k" to avoid confusion with "j")

            Call the k that works "k1" (by EI)
            Now show (2k1+1)+1 is even:

              Use algebra: (2k1+1)+1 = 2k1+2 = 2(k1+1)
            So, take j = k1+1
            i.e., ∃j[c+1=2j], by EG, namely: j = k1+1
            c+1 is even,
            i.e., Even(c+1).
          Now that we know Even(c+1) on the assumption that Odd(c),
          we also know Odd(c) → Even(c+1)
        ∴ we can infer (∀ integer n)[Odd(n) → Even(n+1)], by UG

    2. Here is a more formal ("cleaned up") version:

        Show: (∀ integer n)[Odd(n) → Even(n+1)]
        Let c be an arbitrary integer
        Show Odd(c) → Even(c+1):

        1. Odd(c) :  temporary assumption for Direct Proof
        2. k[c=2k+1] :  1; def of Odd
        3. c=2k1+1 :  2; EI
        4. c+1=2k1+1+1 :  3; algebra
        …(cue the closing music and a deep voice: "to be continued next time")

Next lecture…

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