Day 1, Mat385: Welcome and First Thoughts

• They will be relatively wide open. You may want to find a partner to work with (at most two -- no groups of three).
• You should find some aspect of discrete math that you want to pursue (perhaps one that we don't study in detail, perhaps one you are particularly interested in that we DO study).
• You'll be writing these up to put on a wiki....
• More details, coming soon....

1. Name
2. Year (Senior, etc.)
3. Major
4. Where are you from?
5. Something special about yourself.

We will be covering portions of the following chapters:

• Chapter 1: Logic
  • strip away meaning in order to concentrate on form
  • translate English into the symbolism of logic (formalize what is often left imprecise)
  • logical reasoning cannot help but stand you in good stead
• Chapter 2: Proofs, Recursion, and Analysis of Algorithms
  • Direct versus indirect proofs, especially for proving logical arguments (from chapter 1).
  • Mathematical induction, leading to recursion.
  • Determining the quality of algorithms, by recursion.
• Chapter 3: Sets
  • Elementary aspects of set theory
  • Counting elements of sets: why some infinities are bigger than others.
  • The similarity between logic and set theory will be more apparent later
• Chapter 5: Graphs and Trees
  • Graphs and Trees capture information in many different contexts (e.g. a family tree, a unix file system)
  • How can we manipulate, store, and access information stored in a graph?
• Chapter 6: Graph algorithms
  • shortest path, minimal spanning tree, traversals, etc.
  • Constructing optimal graphs
• Chapter 7: Boolean Algebras and Computer Logic
  • Operations in logic and set theory are specific cases of Boolean Algebras
  • There is a nice correspondence between boolean expressions, truth functions (or tables), and logic networks (which carry out logical functions for technological applications)
  • We simplify, or optimize, logic networks
  • Binary arithmetic is a particular special application of this section
• Chapter 8: Finite State Machines
  • Regular expressions
  • We build machines that recognize expressions (strings of characters), and optimize the machines.

• Charles Dodgson (Carroll's mathematical alter ego, and creator) considered symbolic logic "as an entertainment for children to develop their powers of logical thought."

• Elements of The Game of Logic
  • Fundamentals: these you will encounter in section 1.1
    • The universe: The set of all objects under discussion
    • Proposition: a statement with truth value
    • A syllogism: a group of statements (called propositions, or premises) that lead to a conclusion
      

      An example syllogism:

      1. Babies are crazy;
      2. Crazy people should be in an asylum.

      What is the logical conclusion, assuming that the premises are true?

      The method: eliminating the common term

      So, in the baby example, the common term "crazy" is eliminated.

    • Here's a classic:
      1. Aristotle is a man.
      2. All men are mortal.
      3. Aristotle is mortal.

    • Three types of propositions:
      1. Some cakes are nice
      2. No cakes are nice
      3. All cakes are nice
      (we'll characterize these as modifiers -- some, no, all -- as "quantifiers").

      Some statements come with "implied quantifiers" (real speech can be very irritating that way!): for example, in the example above, "Babies are crazy" should be interpreted to mean "All babies are crazy." There's an implied universal ("all") quantifier.

      Other statements need to be re-written to look like our three types of propositions: e.g. "None but the brave deserve the fair" is re-written as "No not-brave persons are persons that deserve the fair."

    • We are often called upon to negate statements, and must know how to move between types of statements (e.g. "not(Some cakes are nice)" is the same logically as "No cakes are nice", for example).

    • The board
      • Bilateral (can be Trilateral, etc., if there are more factors -- e.g. "cakes", "nice", "chocolate")
      • Counters:

        Here's Carroll's description of the counters:

        Let us agree that a Red Counter, placed within a Cell, shall mean "This Cell is occupied" (i.e. "There is at least one Thing in it").

        Let us also agree that a Red Counter, placed on the partition between two Cells, shall mean "The Compartment, made up of these two Cells, is occupied; but it is not known whereabouts, in it, its occupants are." Hence it may be understood to mean "At least one of these two Cells is occupied: possibly both are."

        Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mind which of two political parties he will join: such a man is said to be "sitting on the fence." This phrase exactly describes the condition of the Red Counter.

        Let us also agree that a Grey Counter, placed within a Cell, shall mean "This Cell is empty" (i.e. "There is nothing in it").

  • Let's see an example worked out:
    • Example 1:
      1. Babies are crazy;
      2. Crazy people should be in an asylum.
      The big question is "What can we logically conclude?"
      • What is the universe?
      • Quantifiers?
      • Descriptors, or factors?
      • Let's play!
        • Create table
        • Use each statement to place (or move) markers.
        • One is "on the fence"!
        • Conclusion?

    • Example 2: Here's another example, written out by Carroll himself. Here it is:

      "That story of yours, about your once meeting the sea-serpent, always sets me off yawning; I never yawn, unless when I'm listening to something totally devoid of interest."

      Let's break it down:

      1. All tellings-of-stories-of-yours-about-once-meeting-the-sea-serpent leave me yawning. (Therefore "No tellings-of-stories-of-yours-about-once-meeting-the-sea-serpent leave me not yawning.")
      2. All yawnings are the result of tellings-of-stories devoid of interest. (Therefore "No yawnings are the result of tellings-of-stories not devoid of interest.")

      The conclusion: "That story of yours, about your once meeting the sea-serpent, is totally devoid of interest."

      If you'd like to try some more, here are a bunch of Carroll's trios to test.