Discrete Structures

Lecture Notes, 15 Sep 2010

Last Update: 15 September 2010

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  1. Propositional Equivalences:

    1. There are a whole host of famous propositional equivalences;
      see the charts on pp. 24–25.

    2. Among the most important are:

      • (AB) ≡ (¬AB) ≡ (¬B → ¬A) ≡ ¬(A ∧ ¬B)
      • the distributive laws (look them up!)
      • De Morgan's Laws (look them up!)

      Construct truth tables to convince yourself of these!

    3. Also note that, because (AB) ≡ (¬AB),
      it's also the case that ¬(AB) ≡ ¬(¬AB)
        (You should convince yourself of this by constructing another truth table!)

  2. Syntactic vs. Semantic Proofs:

    1. There are 2 different ways of proving things:

      1. semantically (in terms of truth values)
      2. syntactically (by means of symbol manipulation)

        • This is the way that computers would have to do it—by manipulating symbols,
          which is what they do best

    2. Here's a theorem for which I'll give 2 proofs, one semantic, one syntactic:

    3. Theorem:
        ¬(AB) ≡ (A ∧ ¬B)

      proof #1 (semantic, using the def of logical equiv):

        Show: For all rows of their t-tables, tval(¬(A → B)) = tval(A ∧ ¬B)

        A B (AB) ¬(A → B) ¬B (A ∧ ¬B)
        T T T F F F
        T F F T T T
        F T T F F F
        F F T F T F

        Because the 4th and 6th columns are identical,
        we can conclude that the 2 propositions are logically equivalent.

      proof #2 (syntactic, using symbol manipulation):

      Note that this symbol manipulation is a little bit like the Transformer toys that turn trucks into monsters, and vice versa.

      Here, we "transformed" ¬(AB) into (A ∧ ¬B)
      by manipulating the symbols according to the rules of Shakespeare's Law and propositional equivalences.

      A semantic "proof" would be needed to decide the truth value of "John gave a book to Mary".

        We need to look at the world to see if that sentence is true.

      But we only need a syntactic "proof" to decide that "John gave a book to Mary" means the same thing as "Mary received a book from John".

        We can determine that they are "equivalent in meaning" just by looking at the syntax of the sentences;
        we don't have to look at the world

      Clearly, you could write a computer program to do this.
      (For more info, see "Automated Theorem Proving")

  3. §1.3: Predicates & Quantifiers

    1. The language of propositional logic is very "coarse grained"

      1. It allows us to talk about The True and The False
        but not about objects and their properties & relations

      2. E.g., can't show the relationship between:

        • My Mac is a networked computer.
        • Some Macs are networked computers.
        • All Macs are networked computers.

        In propositional logic, these would be represented by p, q, r

    2. So we need an "object-oriented", finer-grained logical language

      • It's called first-order predicate logic (or sometimes "first-order predicate calculus",
        which has nothing to do with MTH 141)

      • We'll call it "first-order logic", or FOL, for short.
Next lecture…

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