Day 4: mat385

Last time

Next time

Today:

• Announcements:
  • Reminder: turn off your rectangles.

  • Roll.

  • You have a problem set due today, and another due Wednesday. Keep up!:)

    Questions about the homework?

  • Your first homework is returned.

    • Graded 9, 17, and 48 (chosen randomly):

      • 9: two correct answers

      • 17: just read correctly -- I appreciate those who tried to make more elegant English statements that were equivalent!

        However, don't forget order of precedence: if you said

        "Violets are not blue or roses are red only if sugar is sweet."
        then order of precedence would have us interpret that as
        "(Violets are not blue or roses are red) only if (sugar is sweet)."
        

        Whoops! Dang our language....

      • 48: some of you didn't get the idea of the hint.

        Having proven (in 47) that $\land$ and $'$ together generate all five connectives, you simply show that $\lor$ and $'$ generate $\land$ (e.g. via truth table).

        Alternatively, you can replicate #47 (producing $\lor$ and $\longrightarrow$ using $\lor$ and $'$ -- twice as much work!).

        One of you proved that $A\land B \iff (A \longrightarrow B')'$: how would you like to say that "(I only if not my brother) -- Not" instead of "my brother and I"....? :)

    • Each homework problem is graded out of 2 points, for a total of 6. I'm always nervous when problems at the end of a problem set are chosen, because there are those that simply don't finish....

    • Make sure that you check your answers in the back of the book when possible! That's legit. And, again, you can always come and look over the solutions manual.

    • You may have a "~" on your paper -- that doesn't cost anything, except spiritually. :) It's a little slap on the wrist. We're not allowed to slap on the wrist anymore, so....

• Last time: we talked about quantifiers and predicate wffs: Quantifiers, Predicates, and Validity

  Here's our mocked up version.

• Today: after finishing off 1.3, we'll begin Highlights of Section 1.4: Predicate Logic

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."