Lecture 17 - Notes
No new Goals for today.
Announcements
- Week 3 Feedback themes:
- Labs are challenging
- Many would like to see me do more examples in class
- Midterm exam details:
- In class on Friday November 1st
- Covers material through today (through BoP Chapter
10; up to and including proof by induction).
- You can bring one double-sided 8.5x11” sheet of hand-written
notes.
- Resources:
- I have made a study
guide of sorts, which lists of the Goals for each lecture; my goal
is for the exam to be a fair assessment of the extent to which you’ve
achieved these outcomes.
- I have provided some Racket
practice problems. You can expect the Racket questions on the exam
to resemble these.
- These are also linked from the course webpage Schedule table, on the
day of the Midterm Exam.
Induction
Example Prove the following proposition: \[
\forall n \in \mathbb{N}, \left( \sum_{i=0}^n 2^i \right) = 2^{n+1} - 1
\]
Proof by Smallest
Counterexample
This is a combination of proof by induction and proof by
contradiction. The basic idea is to show the base case holds, then prove
the inductive hypothesis by contradiction. You can do this by supposing
that it does not always hold that \(S_{k-1} \Rightarrow S_{k}\). We let \(k\) be the smallest example for which this
is the case, and thus suppose \(S_{k-1}\) is true but \(S_k\) is not; then show that this leads to
a contradiction.
Example
Proposition: If \(n \in
\mathbb{N}\), then \(4 \mid (5^n -
1)\).
This can be proved by smallest counterexample. See proof BoP
section 10.3, p191.