lect_28 - Structural induction Counting I Margaret M. Fleck...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Structural induction Counting I Margaret M. Fleck 5 April 2010 This lecture finishes structural induction (in section 4.3 in Rosen) and starts the topic of counting, covering sections 5.1 and 5.3. 1 Announcements Midterm coming up on Wednesday. If you have a conflict, you should have received email from me with the details of the conflict exam. If not, please contact me right away. 2 Structural induction with 2D points Last class, we saw examples of structural induction proofs using trees. Structural induction is used to prove a claim about a set T of objects which is defined recursively. Instead of having an explicit induction variable n , our proof follows the structure of the recursive definition. Show the claim holds for the base case(s) of the definition of T For the recursive cases of T s definition, show that if the claim holds for the smaller/input objects, then it holds for the larger/output objects. 1 To see how this works on a set of things that arent trees, consider the following recursive definition of a set S of 2D points: 1. (3 , 5) S 2. If ( x,y ) S, then ( x + 2 ,y ) S 3. If ( x,y ) S, then (- x,y ) S 4. If ( x,y ) S, then ( y,x ) S Whats in S ? Starting with the pair specified in the base case (3 , 5), we use rule 3 to add (- 3 , 5). Rule 2 then allows us to add (- 1 , 5) and then (1 , 5). If we apply rule 2 repeatedly, we see that S contains all pairs of the form (2 n + 1 , 5) where x is a natural number. By rule 4, (5 , 2 n + 1) must also be in S for every natural number n . We then apply rules 2 and 3 in the same way, to show that (2 m +1 , 2 n +1) is in S for every natural numbers m and n ....
View Full Document

This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.

Page1 / 6

lect_28 - Structural induction Counting I Margaret M. Fleck...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online