Download presentation

Presentation is loading. Please wait.

Published byLilly Lowery Modified over 3 years ago

1
CSE 20 DISCRETE MATH Prof. Shachar Lovett http://cseweb.ucsd.edu/classes/wi15/cse20-a/ Clicker frequency: CA

2
Todays topics Proof by contraposition Proof by cases Section 3.5 in Jenkyns, Stephenson

3
Proof by contraposition To prove a statement of the form You can equivalently prove its contra-positive form Remember: (p→q) ( q→ p)

4
Truth table for implication pqp → q TTT TFF FTT FFT Rule this row out!

5
Contrapositive proof of p→q Procedure: 1. Derive contrpositive form: ( q→ p) 2. Assume q is false (take it as “given”) 3. Show that p logically follows

6
Example Thm.: “Let x,y be numbers such that x 0. Then either x+y 0 or x-y 0.” Proof: Given (contrapositive form): Let WTS (contrapositive form): … Conclusion: … ???

7
Example Thm.: “Let x,y be numbers such that x 0. Then either x+y 0 or x-y 0.” Proof: Given (contrapositive form): Let … A. x+y 0 or x-y 0 B. x+y=0 or x-y=0 C. x+y=0 and x-y=0 D. y 0 E. None/more/other

8
Example Thm.: “Let x,y be numbers such that x 0. Then either x+y 0 or x-y 0.” Proof: Given (contrapositive form): Let x+y=0 and x-y=0 WTS (contrapositive form): A. x 0 B. x=0 C. x+y 0 or x-y 0 D. x+y=0 or x-y=0 E. None/more/other

9
Example Thm.: “Let x,y be numbers such that x 0. Then either x+y 0 or x-y 0.” Proof: Given (contrapositive form): Let x+y=0 and x-y=0 WTS (contrapositive form): x=0 Conclusion: … Try it yourself first

10
When should you use contra-positive proofs? You want to prove Which is equivalent to So, it shouldn’t matter which one to prove In practice, one form is usually easier to prove - depending which assumption gives more information (either P(x) or Q(x))

11
Breaking a proof into cases Sometimes it is useful to break a proof into a few cases, and then prove each one individually We will see an example demonstrating this principle 11

12
Breaking a proof into cases 6 people at a party Any two people either know each other, or not (it is symmetric: if A knows B then B knows A) Theorem: among any 6 people, either there are 3 who all know each other (3 friends), or there are 3 who all don’t know each other (3 strangers) 12

13
Breaking a proof into cases Theorem: Every 6 people includes 3 friends or 3 strangers Proof: The proof is by case analysis. Let x denote one of the 6 people. There are two cases: Case 1: x knows at least 3 of the other 5 people Case 2: x knows at most 2 of the other 5 people Notice it says “there are two cases” You’d better be right there are no more cases! Cases must completely cover possibilities Tip: you don’t need to worry about trying to make the cases “equal size” or scope Sometimes 99% of the possibilities are in one case, and 1% are in the other Whatever makes it easier to do each proof

14
Breaking a proof into cases Theorem: Every 6 people includes 3 friends or 3 strangers Case 1: suppose x knows at least 3 other people. Case 1.1: No pair among these 3 people know each other. Case 1.2: Some pair among these people know each other. Notice it says: “This case splits into two subcases” Again, you’d better be right there are no more than these two! Subcases must completely cover the possibilities within the case

15
Breaking a proof into cases Theorem: Every 6 people includes 3 friends or 3 strangers Case 1: suppose x knows at least 3 other people Case 1.1: No pair among these people know each other. Case 1.2: Some pair among these people know each other. Proof for case 1.1: Let y,z,w be 3 friends of x. By assumption, none of them knows each other. So {y,z,w} is a set of 3 strangers.

16
Breaking a proof into cases Theorem: Every 6 people includes 3 friends or 3 strangers Case 1: suppose x knows at least 3 other people Case 1.1: No pair among these people know each other. Case 1.2: Some pair among these people know each other. Proof for case 1.2: Let y,z be 2 friends of x who know each other. So {x,y,z} is a set of 3 friends.

17
Breaking a proof into cases Theorem: Every 6 people includes 3 friends or 3 strangers Case 2: suppose x knows at most 2 of the other 5 people So, there are at least 3 people x doesn’t know Cases 2.1: All pairs among these people know each other. Case 2.2: Some pair among these people don’t know each other. Then this pair, together with x, form a group of 3 strangers. So the theorem holds in this subcase.

18
Breaking a proof into cases Theorem: … Proof: There are two cases to consider Case 1: there are two cases to consider Case 1.1: Verify theorem directly Case 1.2: Verify theorem directly Case 2: there are two cases to consider Case 2.1: Verify theorem directly Case 2.2: Verify theorem directly 18

19
Next class Indirect proof techniques Read section 3.5 in Jenkyns, Stephenson

Similar presentations

Presentation is loading. Please wait....

OK

Introduction to Theorem Proving

Introduction to Theorem Proving

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on switching devices on verizon Ppt on tourism in nepal Ppt on turbo generators manufacturers Ppt on carbon dioxide laser Ppt on new technology in mobiles Ppt on job evaluation process Ppt on water activity symbol Ppt on indian textile industries in bangladesh Ppt on bionics Ppt on kingdom monera organisms