Day 6: mat385

Last time

Next time

Today:

• Announcements:
  • Reminder: turn off your rectangles.

  • Roll.

  • You have a problem set due today, and another due Wednesday.

    Questions about the homework?

  • Your 1.3 homework is returned.

    • Graded 9, 21, and 36:

      • 9(a,b): scope.... follow the brackets and parentheses. But sometimes we don't use them (e.g. b), when "the context is clear".

      • 21(a,f): "there exists" with "and"; "for all" with "implication"

      • 36(a,c): both are valid! a, by the equivalence of negation; and c, we can prove using an interesting strategy.

• Today:

  • Last time we hit chapter 2, section 1: Proof Techniques

    I tried to relate proofs to the material of chapter 1 (an ugly proof involving evenness).

    We saw that, if we believe that a theorem is correct (i.e. we can't think of a counter-example), then there are various approaches to demonstrating it.

    It's always good to have several strategies up your sleeve.

  • Today we cover one of my favorite techniques: Induction

    (and it starts with dominoes!)

Links:

• What does it mean when one says that "The domain is the whole world."? (e.g. p. 51, #13).

  On a set containing all sets: Russell's paradox.

  

  "Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

  "Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox has prompted much work in logic, set theory and the philosophy and foundations of mathematics."