- The Quiz
- Problem 1: first of all, what does it mean for a number to
be divisible by 3? If we don't get that right, then the
proof becomes impossible....
As you can see from the key, students took different
approaches to this.
- Problem 2: This was an example from the text, section 2.1,
Example 7 (p. 103 -- 7th edition).
It asks us to prove the pigeonhole principle (but using
the converse).
- We spent almost two days on induction, and some of you
didn't know how to start....:( So I didn't do a good
job of communicating dominoes!
This was example 18, p. 116 (7th edition).
First Principle:
- Base case: Establish that P(1) is true (the first domino falls)
- Induction step: Assume P(k), and show that P(k+1) follows: i.e., that
\[
P(k) \rightarrow P(k+1)
\]
(if the \(k^{th}\) domino falls, so does the \((k+1)^{th}\)).
Second Principle:
- Base case: Establish that P(1) is true (the first domino falls)
- Induction step: Assume P(r), \(1 \le r \le k\), and show that P(k+1) follows: i.e., that
\[
(\forall r)\left(1\le r\le k \rightarrow P(r)\right) \rightarrow P(k+1)
\]
(if all of the dominoes up to the \(k^{th}\) domino fall, so does the \((k+1)^{th}\)).
- Some tips for induction:
- Algebra on problem 3 caused a problem or two. We need to
know our rules of exponentials.
