Download presentation

Presentation is loading. Please wait.

Published bySavannah Tash Modified over 4 years ago

1
Lecture 2 Introduction To Sets CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine

2
CSCI 1900 Lecture 2 - 2 Lecture Introduction Reading –Rosen - Section 2.1 Set Definition and Notation Set Description and Membership Power Set and Universal Set Venn Diagrams

3
CSCI 1900 Lecture 2 - 3 Set Definition Set: any well-defined collection of objects –The objects are called set members or elements –Well-defined - membership can be verified with a Yes/No answer Three ways to describe a set –Describe in English S is a set containing the letters a through k, inclusively –Roster method - enumerate using { } ‘Curly Braces’ S = {a, b, c, d, e, f, g, h, i, j, k} –Set builder method ; Specify common properties of the members S = { x | x is a lower case letter between a and k, inclusively}

4
CSCI 1900 Lecture 2 - 4 Set Description Examples Star Wars films S = {car, cat, C ++, Java} {a,e,i,o,u,y} The 8 bit ASCII character set Good SciFi Films S = { 1, car, cat, 2.03, …} a,e,i,o,u & sometimes y The capital letters of the alphabet GoodNot So Good

5
CSCI 1900 Lecture 2 - 5 Finite Set Examples Coins –C = {Penney, Nickel, Dime, Quarter, Fifty ‑ Cent, Dollar} Data types –D = {Text, Integer, Real Number} A special set is the empty set, denoted by –Ø –{ }

6
CSCI 1900 Lecture 2 - 6 Infinite Set Examples The set of all integers Z –Z = { …, -3, - 2, -1, 0, 1, 2, 3, …} The set of positive Integers Z + (Counting numbers) – Z + = { 1, 2, 3, …} The set of whole numbers W – W = { 0, 1, 2, 3, …} The Real Numbers R –Any decimal number The Rational Numbers Q –Any number that can be written as a ratio of two integers Example of a number that is in R but not in Q ?

7
CSCI 1900 Lecture 2 - 7 Additional Set Description The set of even numbers E –E is the set containing … -8, -6, -4, -2, 0, 2, 4, 6, 8, … –E = any x that is 2 * some integer –E = Set of all x | x = 2*y where y is an integer –E ={ x | x = 2*y where y is an integer } –E = { x | x = 2*y where y is in Z } –E = { x | x = 2*y where y Z }

8
CSCI 1900 Lecture 2 - 8 Set Membership x is an element of A is written x A –Means that the object x is in the set A x is not an element of A is written x A Given: S={1, -5, 9} and Z + the positive Integers – 1 S 1 Z + – -5 S -5 Z + – 2 S 2 Z +

9
CSCI 1900 Lecture 2 - 9 Set Ordering and Duplicates Order of elements does not matter –{1, 6, 9} = {1, 9, 6} = {6, 9, 1} Repeated elements do not matter –{1, 1, 1, 1, 2, 3} = {1, 2, 3} = {1, 2, 2, 3}

10
CSCI 1900 Lecture 2 - 10 Set Equality Two sets are equal if and only if they have the same elements –S1 = {1, 6, 9} –S2 = {1, 9, 6} –S3 = {1, 6, 9, 6} S1 = S2 - same elements just reordered S2 = S3 - remember duplicates do not change the set Since S1= S2 and S2 = S3 then S1=S3

11
CSCI 1900 Lecture 2 - 11 Subsets A is a subset of B, if and only if every element of set A is an element of set B –Denoted A B Examples –{Kirk, Spock} {Kirk, Spock, Uhura} –{Kirk, Spock} {Kirk, Spock} For any set S, S S is always true –{Kirk, Sulu} {Kirk, Spock, Uhura}

12
CSCI 1900 Lecture 2 - 12 Proper Subsets If every element of set A is an element of set B, AND A≠B then A is a proper subset of B, denoted A B Examples –{1,2} {1,2,3} –{2} {1,2,3} –{3,3,3,1} {1,2,3} –{1,2,3} {1,2,3} But {1,2,3} {1,2,3} –{2,3,1} {1,2,3} But {2,3,1} {1,2,3}

13
CSCI 1900 Lecture 2 - 13 Membership and Subset Exercise Given: D = { 1, 2, {1}, {1,3}} Is 1 D ? Is 3 D ? Is 1 D? Is {1} D ? Is {2} D ? Is {1} D? Is {1} D? Is {3} D? Is { {1} } D ? Is { {1,2} } D ?

14
CSCI 1900 Lecture 2 - 14 Subsets and Equality Given: Two sets A and B –If you know that A B and B A then you can conclude that A = B –If A B then it must be true that B A

15
CSCI 1900 Lecture 2 - 15 Power Set The power set P of a set S is a set containing every possible unique subset of S –Written as P(S) P(S) always includes – S itself –The empty set

16
CSCI 1900 Lecture 2 - 16 Power Set Example

17
CSCI 1900 Lecture 2 - 17 Set Size The cardinality of set S, denoted |S|, is the number distinct elements of S. –if S = {1,3,4,1}, then |S|=3 –|{1,3,3,4,4,1}| = 3 –|{2, 3, {2}, 5} | = 4 –|{ 2, 3, {2,3}, 5, { 2,{2,5} } }| = 5 –|Z | = ∞ –|Ø| = 0 A set is finite if it contains exactly n elements –Otherwise the set is infinite

18
CSCI 1900 Lecture 2 - 18 Universal Set There is no largest set containing everything We will use a (different) Universal Set, U, for each discussion – It is the set of all possible elements of the type we want to discuss, for each particular problem For an example involving even and odd integers we might say U = Z

19
CSCI 1900 Lecture 2 - 19 Venn Diagrams –A graphic way to show sets and subsets, developed by John Venn in the 1880’s –A set is shown as a Circle or Ellipse, and the Universal set as a rectangle or square –This shows that S1 Z, and if x S1 then x Z U = Z S1 = Integers divisible by 2

20
CSCI 1900 Lecture 2 - 20 Venn Diagrams: Subsets U = Z S1 = Integers divisible by 2 This shows that S1 Z and S2 Z and S2 S1 If x S2 then x S1, if x S1 then x Z, if x S2 then x Z S2 = Integers divisible by 4

21
CSCI 1900 Lecture 2 - 21 Venn Diagrams: Subsets 2 U = Z S1 = Integers divisible by 2 S3 = Integers divisible by 5,, This shows that S1 Z and S3 Z, if x S1 then x Z, if x S3 then x Z, and there exists at least one element y such that y Z and y S1 and y S3

22
CSCI 1900 Lecture 2 - 22 Venn Diagram Exercise Draw a Venn Diagram representation for the following example: –U = { x | x W and x < 10 } –A= {1, 3, 5, 7, 9} –B = { 1, 5, 7} –C = {1, 5, 7, 8}

23
CSCI 1900 Lecture 2 - 23 Key Concepts Summary Definition of a set Ways of describing a set Power sets and the Universal set Set Cardinality Draw and interpret Venn Diagrams

Similar presentations

© 2019 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