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Set and Sets Operations

Set and Sets Operations. Set definition. Set is the fundamental discrete structure on which all other discrete structures are built. Sets are used to group objects together. Often, the objects in a set have similar properties. A set is an unordered collection of objects .

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Set and Sets Operations

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  1. Set and Sets Operations Lecturer: Shaykhah

  2. Set definition Set is the fundamental discrete structure on which all other discrete structures are built. Sets are used to group objects together. Often, the objects in a set have similar properties. A setis an unordered collection of objects. The objects in a set are called the elements, or members, of the set Lecturer: Shaykhah

  3. Some Important Sets The set of natural numbers: N = {0, 1, 2, 3, . . .} The set of integers: Z = {. . . ,−2,−1, 0, 1, 2, . . .} The set of positive integers: Z+ = {1, 2, 3, . . .} The set of fractions: Q = {0,½, –½, –5, 78/13,…} Q ={p/q | pЄZ , qЄZ, and q≠0 } The set of Real: R = {–3/2,0,e,π2,sqrt(5),…} Lecturer: Shaykhah

  4. Notation used to describe membership in sets • a set A is a collection of elements. • If x is an element of A, we write xA; If not: xA. • xASay: “x is a member of A” or “x is in A”. • Note: Lowercase letters are used for elements, capitals for sets. • Two sets are equal if and only if they have the same elements A= B : x( x A  x B) also • Two sets A and B are equal if A  B and B  A. • So to show equality of sets A and B, show: • A  B • B  A Lecturer: Shaykhah

  5. How to describe a set? • List all the members of a set, when this is possible. We use a notation where all members of the set are listed between braces. { } Example : {dog, cat, horse} • Sometimes the brace notation is used to describe a set without listing all its members. Some members of the set are listed, and then ellipses (. . .) are used when the general pattern of the elements is obvious. Example: The set of positive integers less than 100 can be denoted by {1, 2, 3, . . . , 99}. Lecturer: Shaykhah

  6. How to describe a set? • Another way to describe a set is to use set builder notation. We characterize all those elements in the set by stating the property or properties they must have to be members. • Example: the set Oof all odd positive integers less than 10 can be written as: O= {x | x is an odd positive integer <10} or, specifying the universe as the set of positive integers, as O = {x  Z+ | x is odd and x<10}. Lecturer: Shaykhah

  7. Sets • The Empty Set (Null Set) • We use to denote the empty set, i.e. the set with no elements. • For example: • the set of all positive integers that are greater than their squares is the null set. • Singleton set • A set with one element is called a singleton set. Lecturer: Shaykhah

  8. Sets • Computer Science • Note that the concept of a datatype, or type, in computer science is built upon the concept of a set. In particular, a datatype is the name of a set, together with a set of operations that can be performed on objects from that set. • For example, Booleanis the name of the set {0, 1} together with operators on one or more elements of this set, such as AND, OR, and NOT. Lecturer: Shaykhah

  9. Computer Representation of Sets • Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The bit string (of length |U| = 10) that represents the set A = {1, 3, 5, 6, 9} has a one in the first, third, fifth, sixth, and ninth position, and zero elsewhere. It is 1 0 1 0 1 1 0 0 1 0. (c)2001-2003, Michael P. Frank

  10. Sets Venn diagrams • Sets can be represented graphically using Venn diagrams. • In Venn diagramsthe universalset U, which contains all the objects under consideration, is represented by a rectangle. • Inside this rectangle, circles or other geometrical figures are used to represent sets. •  Sometimes points are used to represent the particular elementsof the set. Lecturer: Shaykhah

  11. Sets Example: • A Venn diagram that represents V = {a, e, i, o, u}, the set of vowels in the English alphabet Lecturer: Shaykhah

  12. Subset • The setAis said to be a subset of B if and only if every element of A is also an element of B. • We use the notation A  Bto indicate that A is a subset of the set B. • We see that A  B if and only if the quantification x (x  A → x  B) is true. Lecturer: Shaykhah

  13. Subsets • For every set S, 1.  S 2. S  S • Proper subset: When a set A is a subset of a set B but A ≠ B, A  B, and A  B. We write A  B and say that A is a proper subset of B • For A  B to be true, it must be the case that x ((x  A)  (x  B)) x ((x  B)  (x  A)) Lecturer: Shaykhah

  14. Subsets Quick examples: • {1,2,3}  {1,2,3,4,5} • {1,2,3}  {1,2,3,4,5} • Is   {1,2,3}? • Is   {1,2,3}? • Is   {,1,2,3}? • Is   {,1,2,3}? Because to conclude it isn’t a subset we have to find an element in the null set that is not in the set {1,2,3}. Which is not the case Yes! No! Yes! Yes! Lecturer: Shaykhah

  15. Subsets Quiz Time: • Is {x}  {x,{x}}? • Is {x}  {x,{x}}? • Is {x}  {x}? • Is {x}  {x}? Yes! Yes! Yes! No! Lecturer: Shaykhah

  16. Finite and Infinite Sets Finite set • Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that nis the cardinality of S. • The cardinality of S is denoted by |S|. • |A B| = |A| + |B| - |A  B| Infinite set A set is said to be infiniteif it is not finite. For example, the set of positive integers is infinite. Lecturer: Shaykhah

  17. Cardinality Find • S = {1,2,3}, • S = {3,3,3,3,3}, • S = , • S = { , {}, {,{}} }, • S = {0,1,2,3,…}, |S| is infinite |S| = 3. |S| = 1. |S| = 0. |S| = 3. Lecturer: Shaykhah

  18. Sets Ways to Define Sets: • Explicitly: {John, Paul, George, Ringo} • Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…} • Set builder: { x : x is prime }, { x | x is odd }. In general { x : P(x) is true }, where P(x) is some description of the set. Lecturer: Shaykhah

  19. The power of a set • Many problems involve testing all combinations of elements of a set to see if they satisfy some property. • To consider all such combinations of elements of a set S, we build a new set that has as its members all the subsets of S. • Given a set S, the power set of Sis the set of all subsets of the set S. The power set of S is denoted by P(S). • if a set has n elements , then the power has 2n elements Lecturer: Shaykhah

  20. The power of a set Example: What is the power set of the set {0, 1, 2}? P({0,1,2}) is the set of all subsets of {0, 1, 2} P({0,1,2})= { , {0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}} What is the power set of the empty set? What is the power set of the set {} P()= {} P({})= {,{}} N.B. the power set of any subset has at least two elements The null set and the set itself Lecturer: Shaykhah

  21. The Power Set Quick Quiz: Find the power set of the following: • S = {a}, • S = {a,b}, • S = , • S = {,{}}, P(S)= {, {a}}. P(S) = {, {a}, {b}, {a,b}}. P(S) = {}. P(S)= {, {}, {{}}, {,{}}}. Lecturer: Shaykhah

  22. Cartesian Products • The order of elements in a collection is often important. • Because sets are unordered, a different structure is needed to represent ordered collections. • This is provided by ordered n-tuples. • The ordered n-tuple (a1, a2, . . . , an) is the ordered collection that has a1 as its first element, a2 as its second element, . . . , and an as its nth element. Lecturer: Shaykhah

  23. Cartesian Products • Let A and B be sets. The Cartesian product of A and B, denoted by A×B, is the set of all ordered pairs (a, b), where aA and bB. A×B = {(a, b) | a  A  b  B}. A1×A2×…×An= {(a1, a2,…, an) | aiAifor i=1,2,…,n}. • A×B not equal to B×A Lecturer: Shaykhah

  24. Cartesian Products Example: What is the Cartesian product A × B × C, where A = {0, 1}, B = {1, 2}, and C = {0, 1, 2}? • Solution: • AxBxC = {(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1), (0,2,2), (1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)} Lecturer: Shaykhah

  25. Notation with Quantifiers • Whenever we wrote xP(x) or xP(x), we specified the universe of using explicit English language • Now we can simplify things using set notation! • Example • xR (x20) •  xZ (x2=1) • Also mixing quantifiers: a,b,cR  xC(ax2+bx+c=0) Lecturer: Shaykhah

  26. Sets Operations Lecturer: Shaykhah

  27. B A UNION The union of two sets A and B is: A  B = { x : x  A v x  B} • If A = {1, 2, 3}, and B = {2, 4}, then • A  B = {1,2,3,4} Lecturer: Shaykhah

  28. B A Intersection The intersection of two sets A and B is: A  B = { x : x  A  x  B} • If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then • A  B = {Lucy} Lecturer: Shaykhah

  29. B A Intersection If A = {x : x is a US president}, and B = {x : x is deceased}, then A  B = {x : x is a deceased US president} Lecturer: Shaykhah

  30. B A Disjoint If A = {x : x is a US president}, and B = {x : x is in this room}, then A  B = {x : x is a US president in this room} =  Sets whose intersection is empty are called disjoint sets Lecturer: Shaykhah

  31. A Complement The complement of a set A is: A = A’ = { x : x  A} If A = {x : x is bored}, then A = {x : x is not bored} =  U • = U and U =  A  B = B  A Lecturer: Shaykhah

  32. Example Let A and B are two subsets of a set E such that AB = {1, 2}, |A|= 3, |B| = 4, A = {3, 4, 5, 9} and B = {5, 7, 9}. Find the sets A, B and E. E A = {1, 2, 7}, B = {1, 2, 3, 4}, E = {1, 2, 3, 4, 5, 7, 9} A713 B 24 59 (c)2001-2003, Michael P. Frank

  33. U A B Difference The set difference, A - B, is: A - B = { x : x  A  x  B } A - B = A  B Lecturer: Shaykhah

  34. U A B Like “exclusive or” Symmetric Difference The symmetric difference, A  B, is: A  B = { x : (x  A  x  B) v (x  B  x  A)} = (A - B) U (B - A) Lecturer: Shaykhah

  35. Symmetric Difference • Example • Let A = {1,2,3,4,5,6,7} B = {3,4,p,q,r,s} Then we have A U B = {1,2,3,4,5,6,7,p,q,r,s} A  B = {3,4} We get A  B = {1,2,5,6,7,p,q,r,s} Lecturer: Shaykhah

  36. Proving Set Equivalences • Recall that to prove such identity, we must show that: • The left-hand side is a subset of the right-hand side • The right-hand side is a subset of the left-hand side • Then conclude that the two sides are thus equal • The book proves several of the standard set identities. • We will give a couple of different examples here. Lecturer: Shaykhah

  37. Proving Set Equivalences: Example A (1) • Let • A={x|x is even} • B={x|x is a multiple of 3} • C={x|x is a multiple of 6} • Show that AB=C Lecturer: Shaykhah

  38. Proving Set Equivalences: Example A (2) • AB  C:  x AB • x is a multiple of 2 and x is a multiple of 3 • we can write x=2.3.k for some integer k • x=6k for some integer k  x is a multiple of 6 • x  C • CAB:  x C • x is a multiple of 6  x =6k for some integer k • x=2(3k)=3(2k) x is a multiple of 2 and of 3 • x  AB Lecturer: Shaykhah

  39. Proving Set Equivalences: Example B (1) • An alternative prove is to use membership tables where an entry is • 1 if a chosen (but fixed) element is in the set • 0 otherwise • Example: Show that A  B  C = A  B  C Lecturer: Shaykhah

  40. Proving Set Equivalences: Example B (2) • 1 under a set indicates that an element is in the set • If the columns are equivalent, we can conclude that indeed the two sets are equal Lecturer: Shaykhah

  41. Lecturer: Shaykhah

  42. Let’s proof one of the Identities Using a Membership Table A  (B U C) = (A  B)U(A  C) Lecturer: Shaykhah

  43. ANY QUESTIONS??? • Refer to chapter 2 of the book for further reading Lecturer: Shaykhah

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