Day 11: mat385

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• Announcements:
  • I hope that you had a good "Spring Break"!:) Hopefully our other "spring break replacement vacation days" will have more spring in them.

  • Homework 2.2: Graded 4, 31, 50

    We're struggling with the induction proofs, but the biggest problem is simply not putting everything together. Please explain more about how you move from step to step (if it's not obvious to your sister in high school, or the like).

    • #4
      • $P(n)$ is an assertion, not a quantity equal to something. Use ":", not "=", as in $P(n): 1+3+\ldots+\frac{n(n+1)}{2}=\frac{n(n+1)(n+1)}{6}$
      • Don't forget the base case.
      • At some point in your argument you have to actually invoke $P(k)$, the inductive hypothesis.
      • You're trying to prove that $P(k) \rightarrow P(k+1)$: if the $k^{th}$ domino falls, then the $(k+1)^{th}$ also falls.
      • Don't forget a conclusion: since the first domino has fallen, and since, if the $k^{th}$ domino falls, then the $(k+1)^{th}$ also falls; therefore, they all fall down: $P(n)$ for all $n\ge 1$.

    • #31: $P(n)$: $n^2>n+1$.
      1. Demonstrate $P(2)$
      2. Assume $P(k)$; Consider the LHS of $P(k+1)$. At some point you need to invoke $P(k).$ Then take it home, showing that $P(k+1)$. Hence $P(k) \rightarrow P(k+1)$, and you
      3. Conclude $P(n)$ for $n\ge 2$ by the first principle of mathematical induction.

    • #50: $P(n): 7$ divides $13^{n}-6^n$.

      Clearly $P(n)$ is equal to nothing here.... It is an assertion, about a quantity which is a function of $n$ -- that it is divisible by 7.

      Now do the usual, talking as you go....

• Today: we'll hit the highlights of Section 3.3: Analysis of Algorithms.
  • Hitting the highlights, Part I: searches
  • Hitting the highlights, Part II: Euclidean Algorithm (start)
  • Hitting the highlights, Part II: Euclidean Algorithm (fin)
  • The marked up pdf (also to come, when I've finished the second part)

• Links:
  • Nothing new to report.