What is Set? Set Theory Peter Lo Set is any well-defined list, collection, or class of objects. The objects in set can be anything These objects are called the Elements or Members of the set. CS218 Peter Lo 2004 1 CS218 Peter Lo 2004 2 Notation Tabular Form = {a, e, i, o, u} Set-builder Form = {x x is odd} (Do Ex. 1 10) Venn Diagram Venn Diagram is a pictorial representation of sets by set of points in the plane. The niversal Set is represented by the interior of a rectangle, and the other sets are represented by disks lying within the rectangle. CS218 Peter Lo 2004 3 CS218 Peter Lo 2004 4
Type of Sets Finite and Infinite Set Finite Infinites Set niversal Set Subset Proper Subset Null Set Disjoint Set Sets of Sets Power Sets Finite Set is a set consisting specific number of different elements. Let X = {1, 2, 3, 4,. 100}, then X is finite. Infinite Set is a set consisting infinite number of different elements. Let X = {1, 3, 5, 7,..}, then X is infinite. CS218 Peter Lo 2004 5 CS218 Peter Lo 2004 6 niversal Set ll the set under consideration can thought of as subsets or another set called the niversal Set. The universal set is denoted by. Subset is a Subset of if every element of is also an element of. Example: Given Set = {a, b, c, d, e}, Set = {a, d}. is a subset of when each elements of is also an elements of. If is subset of, written. CS218 Peter Lo 2004 7 CS218 Peter Lo 2004 8
Proper Subset is a Proper Subset of if is a subset of and is not equal to. Example: Given Set = {a, b, c, d, e}, Set = {a, d}, Set C = {a, d}. Then is a proper subset of, but is not a proper subset of C. If is proper subset of, written. Null Sets set with no elements is called Empty Set (or Null Set). The Null Set φ is a subset of every set. The Empty Set is denoted by φ. Let = { x x 2 = 4, x is odd}, then = φ. CS218 Peter Lo 2004 9 CS218 Peter Lo 2004 10 Disjoint Sets Two Sets and are called Disjoint Sets if they have nothing in common. Example: Let = {a, b}, = {c, d}, then and are disjoint sets. In Set Notation, = φ. Power Sets The Power Sets of a set X, is denoted by P(X), is the set of all subset of X. If a finite set X has n elements, then power set of X contain 2 n elements. Let Set X = {0, 1}, then P(X) = {φ, {0}, {1}, {0, 1}}. CS218 Peter Lo 2004 11 CS218 Peter Lo 2004 12
Sets of Sets Sets can be the elements of another sets. We called them Sets of Sets, Family of Sets and Class of Sets. The set = {{2, 3}, {2}, {5, 6}} is a family of sets. Its members are the sets {2, 3}, {2}, {5, 6}. Operation of Sets nion Intersection Difference Complement CS218 Peter Lo 2004 13 CS218 Peter Lo 2004 14 nion of Sets If the elements of some set are joined with the elements of some set, the nion of set and set is formed. The union of set and set is denoted as. Intersection of Sets The set of elements that are common to set and set is called Intersection of set and set. The intersection of set and set is denoted as CS218 Peter Lo 2004 15 CS218 Peter Lo 2004 16
Difference The Difference of Set and is the set of elements which belong to but don t belong to. The different of set and set is denoted as, \, ~. Complement The Complement of Set is the Set of elements which do not belong to. The Complement of is denoted by,, c. (Do Ex. 11 16) CS218 Peter Lo 2004 17 CS218 Peter Lo 2004 18 ttributes of Sets Equality of Sets Comparability Cardinality Principle of Inclusion and Exclusion Equality of Sets Set is equal to set is they both contains the same elements. The equality of and is denoted by =. We can also say = if and only if and. CS218 Peter Lo 2004 19 CS218 Peter Lo 2004 20
Comparability Two sets and are said to be Comparable if one of the sets is a subset of the other set. Two set are said to be Incomparable if neither one of the set is a subset of the others. Cardinality The number of elements in a set is called its Cardinality. Cardinality is denoted by. {1, 3, 5, 7, 9} = 5 CS218 Peter Lo 2004 21 CS218 Peter Lo 2004 22 Principle of Inclusion and Exclusion If and be finite sets, then = + - Let = {1, 3, 5}, = {1, 2, 3, 4} Then = 5, = 3, = 4, = 2; + - = 3 + 4 2 = 5. Proofs sing Venn Diagrams (Do Ex. 17 19) Proof using lgebra Laws (Do Ex. 20 21) Three Set Venn Diagram (Do Ex. 22) Venn Diagrams using Regions (Do Ex. 23 30) CS218 Peter Lo 2004 23 CS218 Peter Lo 2004 24