1 / 11

Induction

Induction. The Idea of Induction. Color the integers ≥ 0 0 , 1 , 2 , 3 , 4 , 5 , … I tell you, 0 is red , & any int next to a red integer is red , then you know that all the ints are red !. Induction Rule. Like Dominos…. Example Induction Proof. Let’s prove:.

judyh
Download Presentation

Induction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Induction

  2. The Idea of Induction Color the integers ≥ 0 0, 1, 2, 3, 4, 5, … I tell you, 0isred, & any int next toared integer isred, then you know that all the ints arered!

  3. Induction Rule

  4. Like Dominos…

  5. Example Induction Proof Let’s prove: (for r ≠ 1)

  6. Example Induction Proof Statements inmagenta form a template for inductive proofs: • Proof: (by induction on n) • The induction hypothesis, P(n), is: (for r ≠ 1)

  7. Example Induction Proof Base Case (n = 0): 1 OK!

  8. Example Induction Proof • Inductive Step: Assume P(n)for some n≥0 and prove P(n+1):

  9. Example Induction Proof Now from induction hypothesisP(n) we have so add rn+1to both sides

  10. Example Induction Proof adding rn+1to both sides, This proves P(n+1) completing the proof by induction.

  11. an aside: ellipsis Means you should see a pattern: “” is an ellipsis. • Can lead to confusion (n= 0?) • Sum notation more precise

More Related