Covers: Midterm Review

Transcription

1 Midterm Review 1. These slides and review points found at 2. ring a photo I card: Rocket ard, river's License overs: 4.1 Graphs + uler Paths 4.2 Traveling Salesman + Hamiltonian Paths 2.1 Sets 2.2 Set Theory 2.3 Set Operations 6.1 Number Theory 11.1 Voting Methods 11.2 efects of Voting Methods 11.3 Weighted Voting Systems 2 1 Know the basic vocabulary of the sections. The test will be multiple choice. The test will be like the online HW rather than the lab assignments. Graphs are made up of 2 parts: v and e graph is connected if 4 3 path is a sequence of adjoined edges along a graph n uler graph visits every once. 5 F 6

2 Mnemonic: uler Graph visits every dge exactly once. graph can be traversed if there is an uler path. There is an uler path if 0 or 2 verticies are odd. 7 8 Pop quiz!!! 1) Find an odd vertex. ulerizing is the process of duplicating edges until all verticies are even. odd 2) Is there an uler path? 9 odd 10 Two solutions. Which is more optimal? complete graph has /07/13 12

3 complete graph has every vertex connected to every other vertex. Hamiltonian path goes through each once. 10/07/ Hamiltonian path goes through each vertex once. So a Hamiltonian path might miss edges. circuit (uler or Hamiltonian) is a path that starts and ends at the same vertex. n uler path goes through each edge once. It might use the same vertex multiple times /07/13 16 The weight of an edge is a number assigned to the edge. (Think distance between cities.) graph is a weighted graph if all of its edges have weights What is the weight of a), b), c), - no edge, no weight 17 Traveling Salesman Problem - visit every city (vertex) with least distance (weight) - so least weight Hamiltonian path

4 Pop quiz!!! Hamiltonian Paths visit each exactly once. Representing Sets Set-builder notation: 1) ircuit 2) dge 3) omplete 4) Vertex 5) None of the above 10/07/ Pearson ducation, Inc. ll rights reserved. Section 2.1, Slide { 1, 2, 3, 4, 10 } { x x is positive and even } { University Hall, Snyder Memorial, Gillham Hall, Field House, Rocket Hall, Palmer,... } set with no entries is known as the empty set. It can also be written as The empty set is a subset of every set. It is not an element of every set, but here is an example: = { ob, 12345,, pumpkins } n() = the number of elements in set = { ob, 12345,, pumpkins } n() = = { a, b, c, x, y, z } n() = = { x x is a day of the week } n() = 10/07/13 23 means "is an element of" means "is not an element of" Pop Quiz!!!! = { 1, 2, 3, 4, 5} = { {1}, {2}, {3}, {4}, {5} } 1) 5 is an element of 2) {5} is an element of 3) {5} is a subset of 10/07/13 24

5 Venn iagram. 26 U

6 Order of Operations ( ) parenthesis always done first. ' set complement next. U,, union, intersection, difference last Methods of solving - Venn iagrams - Shorthand ' = U ' = - Longhand U ' = U = { 1, 2, 3, 4, 5, 6 } = { 1, 2, 3, 4 } = { 2, 4, 6 } = { 3, 4, 5 } Pop Quiz!!! How many elements in? Find ( U ) ' 33 a b means a divides b If a b then b = a c where c is some other number. This is a factor of b and 1000 / 20 = 50, so 1000 = 20 x 50 10/07/13 34 number who's factors are only 1 and itself is a prime number. 2, 3, 5, 7, 11, 13, 23, 29, 31, 37, etc factor tree splits a number into 2 factors at each step. xample: 420 You can use a Sieve of ratosthenes to find them. 10/07/13 35 The prime factorization is the collection of all the primes. 10/07/13 36

7 G = greatest common divisor = use lowest power of factors LM = least common multiple = use largest power of factors /07/ /07/13 38 There are different voting methods when you have more than 2 candidates. 1) Plurality 2) orda ount 3) Plurality with limination 4) Pairwise omparison 10/07/13 39 Plurality Person with most votes wins. orda ount dd up points for each candidate. The one with the most points wins. On a ballot, a last place vote gets 1 point, second to last place vote gets 2 points, etc. 10/07/13 40 xample 3 candidates, 5 voters. Voter 1: Voter 2: Voter 3: Voter 4: Voter 5: Plurality with limination Remove candidate with least 1 st place votes. Retally ballots. Repeat until 1 person remains. This person is the winner. 10/07/ /07/13 42

8 xample 3 candidates, 5 voters. Voter 1: Voter 2: Voter 3: Voter 4: Voter 5: Pairwise omparison ompare every pair of candidates. If one wins, they get 1 point, the other 0. If they tie, they both get ½ points. The candidate with the most points wins. 10/07/ /07/13 44 xample 3 candidates, 5 voters. Voter 1: Voter 2: Voter 3: Voter 4: Voter 5: Fairness onditions and riteria for analyzing defects. 10/07/ /07/13 46 Pop quiz!!!!! Which of these is not a voting method? 1) orda ount 2) Plurality with lmination 3) Plurality 4) Majority riterion 5) Pairwise omparison No current voting method satisfies all of these well-meaning conditions and criteria 10/07/ /07/13 48

9 ny voters that vote the same way is called a coalition. xample: winning coalition is a coalition that can always pass an issue / meets the quota. [ 10: 2, 2, 2, 4, 4 ] xample: Find a winning coalition of voters. [ 10: 2, 2, 4, 4, 4 ] Who is critical to get 8 votes? 51 {} 5 {} 3 {} 4 {,} 8 winning {,} 9 winning {,} 7 {,,} 12 winning 52 ompute the anzhaf Power Index for,,. critical {} 5 {} 3 {} 4 {,} 8 winning, {,} 9 winning, {,} 7 {,,} 12 winning ompute the anzhaf Power Index for,,. critical {} 5 {} 3 {} 4 {,} 8 winning, (Note: total {,} 9 winning, critical voters {,} 7 is 5 = 3+1+1) {,,} 12 winning critical 3 times, critical 1 time, critical 1 time anzhaf Power Index : 3/(3+1+1) : 1/(3+1+1) : 1/(3+1+1)