Download presentation

Presentation is loading. Please wait.

Published bySelena Brazier Modified over 2 years ago

1
Lecture 13 3.2,3.3 Sequences & Summations Proof by Induction

2
Sums geometric progression: arithmetic progression: some other useful sums: derivative a=1, n infinity

3
Sums Example: set notation:note: the order doesn’t matter when sum these elements.

4
Cardinality definition: Two sets have the same cardinality if and only if there is a one-to-one correspondence between them. This is simple for finite sets, but what if a set has infinite elements? definition: A set that is finite or has the same cardinality as the set of positive integers (Z+) is called countable. Example: Consider the sequence {an}, an = n^2, n={1,2,3,4...} Naively speaking, there seem to be much less elements in {an} than in Z+ (since we skip a lot). Infinity is weird! Here is the one-to-one mapping: 1 2 3 4 5 6 7... 1 4 9 16 25 36 49... infinity(intuitively: you can enumerate them)

5
cardinality Now what about the positive rational numbers: p/q with p,q integer, q not 0 ? 1 2 3 4 5 6 7 8 9 10 1 ½ 2 3 1/3 ¼ 2/3 3/2 4 5 skip set of positive rational numbers are countable, in fact the set of all rational numbers is countable : 4 countable quadrants.

6
cardinality That was amazing? Now check this out: a proof that the real numbers are not countable! When a prove seems impossible, try to prove a contradiction. First this: unions of countable sets are countable intersections of countable sets are countable a subset of a countable set is countable. Prove: assume that the reals are countable, the reals in (0,1) are countable (since it is a subset), Thus there is a sequence as follows: r1 = 0.d11 d12 d12 d14... r2 = 0.d21 d22 d23 d24... r3 = 0.d31 d32 d33 d34... etc. construct the number: r = 0.c1 c2 c3 c4... with ci=4 if dii not 4 ci = 5 if dii = 4 This is guaranteed to be different than any real in the list, so it isn’t in the list, so the list is not complete. contradiction! real in [0,1] are uncountable all reals are uncountable.

7
3.3 Mathematical Induction If we want to prove propositions P(k) for all positive integers, we may use inductions. First we prove: P(1) is true. Then we prove P(k) P(k+1). So, is P(100) true? yes, use “modus ponens” 99 times. P(1) P(1) P(2) ____________ P(2) P(2) P(3) ___________ P(3) until P(100). formally:

8
Examples prove that the sum of the first n odd positive integers is n^2. n=1: 1 = 1. assume it’s true for some k. 1+3+5+...+2k-1 = k^2 is true. add 2k+1 on each side: 1+3+5+...+2k-1 + 2k+1= k^2 + 2k + 1 1+3+5+...+2(k+1)-1 = (k+1)^2

9
Examples Prove that n < 2^n for positive integers n. P(1): 1 < 2 Inductive step: assume P(k) is correct, prove P(k+1) is correct. k < 2^k k+1 < 2^k + 1 < 2^k + 2^k = 2^(k+1) Note: we could of course also start our induction at another integer b. Example: Prove that 2^n = 4. P(4) = 2^4 = 16 < 4! = 4 3 2 1 = 24. Induction: assume 2^k < k! 2 x 2^k < 2 x k! 2^(k+1) = 4). 2^(k+1) < (k+1)!

10
Fun Example Show that a chessboard with 2^n x 2^n squares where one arbitrary square has been removed can be tiled with L-shapes. (needs drawing) P(1): All 4 possibilities of removing the square for a 2x2 example are precisely covered with 1 L-shape. Assume P(k) is true. Now construct a chessboard with that is twice as large in both directions. Equally divide it into 4 pieces. Remove one piece arbitrarily from one of the 4 pieces. Since P(k) is true that piece can be covered with L-shapes. Next place one L-shape in the middle to remove one square from the remaining 3 pieces. Again due to P(k) these can now be covered as well.

11
Strong Induction induction (I): strong induction (SI): SI I: I SI: they are equivalent

12
Strong Induction Example: Consider the game where there are 2 piles of n matches. Each player picks an arbitrary number of matches from one pile. The one who gets the last matches wins. Proposition: The player who starts second can always win. P(1): 2 piles with 1 match each. Second player always wins. Assume player 2 wins when we have 2 piles of k matches. Can player 2 win when we have 2 piles of k+1 matches? Player 1 will have to take between 1 and k+1 matches from 1 pile. If he picks k+1 matches, player 2 wins by grabbing all matches from the other pile. If he picks between 1 and k matches, player two takes the same amount from the other pile reducing the problem to a smaller equivalent problem. Strong Induction player 2 always wins

13
Examples Proposition: every positive integer n>2 can be written as the product of primes. P(2): product of itself (it’s prime). Assume k can be written as a product of primes. Can we prove it for k+1? two cases: k+1 = prime (thus it is a product of one number – itself). k+1 = a x b However since both a and b = 2, we know that a and b can be written as the product of primes a x b is a product of primes.

14
The well ordered property Every non-empty set of nonnegative integers has a least element. This is a trivial statement made explicit so we can give it a name in a proof... Example: Round-Robin tournament: n players play against each other. A cycle is a situation where p1 beats p1, p2 beats p3, pn beats p1. Proposition: If there is a cycle of length greater than 3, then there is also a cycle of 3 among the people in the larger cycle. Prove: (by contradiction) Assume that there is a cycle of length k, where k is the smallest integer > 3 for which a cycle exist and no cycle of length 3 exists. Cycle: p1 p2 p3... pk k> 3 Look at p1 p2 p3 if p3 beats p1 we have a cycle of length 3 (contradiction) if p1 beat p3 we can construct the cycle p1 p3 p4... pk which also leads to a contradiction.

15
Infinite Descent A method to prove that a propositional function P(k) is false for all positive integers k. Assume that P(k) is true for at least one k. By the well ordered property there must be a least element s, such that P(s) is true. We show that there is a s’ for which P(s’) is true with s’

Similar presentations

Presentation is loading. Please wait....

OK

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on disaster recovery plan Ppt on revolution of the earth and seasons kids Ppt on molecular biology of head neck cancer Ppt on ledger and trial balance Ppt on bluetooth communication ios Ppt on astronomy and astrophysics encyclopedia Ppt on tcp ip protocol suite 3rd Ppt on net etiquettes Ppt on sectors of indian economy Ppt on nutrition in plants for class 7