Converse, inverse, and contrapositive. Example given of
CAPalert,
where it is claimed of movies that
"If it's inappropriate for children, it's inappropriate for adults!"
So therefore, if a movie is appropriate for adults, it's appropriate
for children?
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Friday
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August 30th
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Starting proofs; proving by breaking into cases, then handling them
one by one.
Three examples given of breaking into cases: a geometric theorem,
a baby number theory one, and one about gluing puzzle pieces together.
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Wednesday
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September 4th
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Proofs by contradiction and by induction. Why does induction work?
It's really because every subset of the natural numbers has a
smallest element.
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Friday
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September 6th
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A sample proof by contradiction: Euclid's proof of the infinitude of primes.
Two sample proofs of induction: the formula for Fibonacci numbers,
and the averaging theorem for juggling patterns.
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Monday
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September 9th
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Strong induction, and the end of the juggling-averaging proof.
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Wednesday
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September 11th
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One-to-one and onto stuff (from section 1.6).
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Friday
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September 13th
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The number of functions from X to Y, and the number of 1:1 functions.
Euclid's algorithm. gcd(m,n) can be written as xm+yn for some x,y. (2.4)
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Monday
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September 16th
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Unique factorization of integers, and its failure in some other contexts.
Congruences. (2.5)
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Wednesday
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September 18th
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More on congruences, division, Fermat's Little Theorem.
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Friday
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September 20th
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Chinese Remainder Theorem.
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Monday
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September 23rd
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RSA encryption.
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Wednesday
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September 25th
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Beginning counting. Sums vs. products. N choose K.
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Friday
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September 27th
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Counting juggling patterns. The recurrence relation on
binomial coefficients, and Pascal's triangle.
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Monday and Wednesday
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September 30th, October 2nd
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Review for midterm #1.
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Friday
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October 4th
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Midterm #1.
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Monday
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October 7th
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More Pascal's triangle; a couple of binomial coefficient identities.
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Wednesday
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October 9th
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The stars-and-bars argument, and multinomial coefficients.
(Section 4.6, Theorems 2 & 3)
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Friday
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October 11th
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The Evil King's scale of justice.
Beginning probability.
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Monday
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October 14th
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The Evil King makes us sort.
Lexicographic order (4.7).
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Wednesday
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October 16th
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Recurrences: Hanoi, Quicksort.
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Friday
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October 18th
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We did 5.1 #17 in class, and came up with
a closed-form solution, but it wasn't as convenient as the actual recurrence!
Also, Catalan numbers.
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Monday
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October 21st
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5.2: Linear recurrences (homogeneous and inhomogeneous).
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Wednesday
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October 23rd
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5.4: Generating functions.
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Friday
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October 25th
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More generating functions. The binomial theorem for noninteger exponents.
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Monday
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October 28th
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Using the binomial theorem to count binary trees.
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Wednesday
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October 30th
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Linear inhomogeneous recurrences. Beginning inclusion-exclusion.
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Friday
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November 1st
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More inclusion-exclusion. Counting derangements.
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M, W, F, W
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November 4th, 6th, 8th, 13th
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Reviewing for, taking, and going over the midterm.
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Friday
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November 15th
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More inclusion-exclusion.
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Monday
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November 18th
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More inclusion-exclusion. The Mobius function.
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Wednesday
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November 20th
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Beginning 1.8: little-o notation. Barely mentioned big-O.
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Friday
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November 22nd
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More on big-O notation.
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