Lecture 2 Introduction To Sets

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

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

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} CSCI 1900

4. 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 Set Description Examples Good Not So Good CSCI 1900

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 • { } CSCI 1900

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 numbersW • 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 ? CSCI 1900

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 } CSCI 1900

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 notan 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+ CSCI 1900

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} CSCI 1900

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 CSCI 1900

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} CSCI 1900

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} CSCI 1900

13. 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 ? Membership and Subset Exercise CSCI 1900

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 CSCI 1900

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  CSCI 1900

16. Power Set Example • Given: S = {x,y,z} • P(S) = {, {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z} } • Nota Bene • If there are n elements in a set S then there are elements in the power set P(S) CSCI 1900

17. Set Size • The cardinalityof 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 CSCI 1900

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 CSCI 1900

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 CSCI 1900

20. Venn Diagrams: Subsets U = Z S1 = Integers divisible by 2 S2 = Integers divisible by 4 • This shows that • S1  Z and S2 Z andS2  S1 • If x  S2 then x S1, if x  S1 then x Z, if x  S2 then x  Z CSCI 1900

21. U = Z S1 = Integers divisible by 2 S3 = Integers divisible by 5 Venn Diagrams: Subsets 2 • This shows that S1 Z and S3 Z,ifx S1 then x Z, if x  S3 then x  Z, and there exists at least one element y such that • y Zand y S1and y S3 CSCI 1900

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} CSCI 1900

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 CSCI 1900