This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Discrete Mathematics Dr. Fred Phelps Lecture 2 More Counting: Permutations and Combinations Announcements • Please register for discrete math on the site http://dl.iitu.kz. The main textbook is on that site. • The syllabus is also on that site. • And a copy of the main textbook: “ Discrete Mathematics” by Lovasz, Pelikan and Veztergombi. LPV 1.3 Number of Subsets • How many subsets does a set of n elements have? • Let • We may include a 1 or not include it in the subset we are building. Thus there are two choices. • This also true for a 2 and so there are 2x2=4 ways we can decide about including or not including the elements { a 1, a 2} in our subset. { } 1 2 1 , ,. .. , . n n A a a a a= Theorem 1.3.1 • A set with n elements has 2 n subsets . • Let’s look at the case of n =3. Y Y Y Y Y Y N N N N N N N Y Eight subsets of A = { a, b, c } Let S be a subset of A Another proof of Theorem 1.3.1 • Every subset of the set { a , b , c } corresponds to one path and only one path down this tree. • There is a question row for each element of A . • Since the number of nodes doubles each time we answer a question, there are subsets of A . 2 A Listing (finite) subsets in a standard order. • One way is to list the empty set, then the subsets with one element, then those with two elements and so on until we have the subset of all the elements as the last subset listed. • We can decide to list the one element subsets in their natural order if there is one (like alphabetical order). Listing (finite) subsets in a standard order. • By this method, the subsets of A = { a , b , c } would be listed as: { } , , , , , , , a b c ab ac bc abc ∅ Another method is the alphabetical listing: { } , , , , , , , . a ab abc ac b bc c ∅ LVP 1.3 Natural Orderings • General Problem: Can we find a way to number the subsets so that we can quickly find, for example, the 23rd subset of a five member set?...
View
Full
Document
This note was uploaded on 10/15/2011 for the course MATHS 100 taught by Professor Fredphelps during the Spring '11 term at Jordan University of Science & Tech.
 Spring '11
 FredPhelps
 Math, Permutations, Counting

Click to edit the document details