Definition of a Set

• DM is study of discrete structures. Fundamental discrete structure upon which most of other discrete structures are built is the sets.

• Unordered collection of objects is called a set. The objects in a set are also called the elements, or members of the set.

• There are several ways to describe a set. One way is to list all members of the set, when possible.

• We have the notation to list elements of a set enclosed between braces and separated from each other by comma.

• Capital letters are used to name sets. • Examples:

– The set of all vowels of English V = { a, e, i, o, u}.– The set O of all odd positive integers less than 10, O = {1,3,5,7,9}.– The set S of all positive integers less than 100, S = {1,2,3,…..99}.

Set Notations• Although sets are usually used to group together elements with

common properties, there is nothing that prevents a set from having apparently unrelated elements.

• For example; V = {a, 2, Fred, New Jersey} is a set containing four elements a, 2, Fred and New Jersey.

• At times the braces notation is used to describe set without listing all of its elements. In such case only some elements are listed and then ellipses i.e. ( … ) are used to indicate general pattern of the elements that are included in the set.

• For Example; set of positive integers less than 100 can be denoted as {1, 2, 3, …, 99}. The set of whole numbers can be represented as {0, 1, 2, 3, … }

• For a set of elements that are infinitely many in descending as well as ascending order, ellipses can be used on either side of listed few elements.

• For Example; set of all integers can be denoted in the form of a set Z = { …, -2, -1, 0, 1, 2, … }.

Set Builder Notation• Set Builder Notation; is another way of describing sets.

Rather than listing we characterize all elements in a set by stating the property or properties that they share.

• For Example; The set O of all positive odd integers less than ten can be written as O = {x | x is an odd positive integer less than ten}.

• Set Builder notation becomes handy in describing sets that have infinitely large elements and can not be listed.

• For example; set of all real numbers can be denoted as R = {x | x is a real number}.

• Similarly set of all rational numbers can be represented as Q = {p/q | p Z, q Z q ≠ 0}.

Equal Sets

• Equal Sets; At times it is mathematically important to assert that two differently specified collection of objects are really the same set or otherwise. It is thus important to understand what it means for two sets to be equal.

• Two sets are equal, if and only if, they have same elements.• For Example; set {1, 3, 5} and set {3, 5, 1} are equal as they

have the same elements 1, 2 and 3. • Note; that the order in which the elements of set are listed does

not matter. Also it does NOT matter if an element of set is listed more than once.

• For example; the two sets {1, 3, 3, 3, 5, 5, 5, 5} & {1, 3, 5} are equal as they have same elements.

Venn Diagrams

• Venn Diagrams; provide another way of representing the sets graphically. In Venn Diagrams the “Universal Set U”, which contains all the objects under consideration is represented by a rectangle’ Inside this rectangle, circles are drawn to represent sets. Some times dots are also used to represent particular element of the set.

• For example; to draw a Venn diagram representing V, set of vowels in English alphabets, we first draw a rectangle to indicate universal set U, which is the set of the 26 letters of English alphabets

• Inside this rectangle, we draw a circle to represent set V (the set of vowels). Inside this circle we indicate the elements of V with dots. It is customary to use capital letters for denoting sets and small case letters to represent elements of set.

Venn Diagram for Set of Vowels