Theory of sets

THEORY OF SETS

We often deal with groups or collection of objects, such a set of books, a group of students, a list of states in a country, a collection of baseball cards, etc. Sets may be thought of as a mathematical way to represent collections or groups of objects. The concept of sets is an essential foundation for various other topics in mathematics. Definition of Sets A set is acollection of objects, things or symbols which are clearly defined and distinct. The individual objects in a set are called the members or elements of the set. A set must be properly defined so that we can find out whether an object is a member of the set. There are two ways of doing this. 1. Listing the elements The set can be defined by listing all its elements, separated by commas and enclosed withinbraces. Example: B = {2, 4, 6, 8, 10} X = {a, b, c, d, e} However, in some instances, it is impossible to list all the elements of a set. In such cases, we define the set by method 2. 2. Describing the elements The set can be defined, where possible, by describing the elements. Example: C = {x : x is an integer, x > – 3 } This is read as: “C is the set of elements x such that x is an integer greaterthan –3.” D= {x: x is a river in a river} We should describe a certain property which all the elements x, in a set, have in common so that we can know whether a particular thing belongs to the set. We relate a member and a set using the symbol ∈. If an object x is an element of set A, we write x ∈ A. If an object z is not an element of set A, we write z ∉ A. ∈ denotes “is an element of‟ or “is amember of” or “belongs to” ∉ denotes “is not an element of” or “is not a member of” or “does not belong to” Example: If A = {1, 3, 5} then 1 ∈ A and 2 ∉ A 3. Venn diagrams. In a Venn diagram, the sets are represented by shapes; usually circles or ovals. The elements of a set are labelled within the circle. Example: Given the set P is the set of even numbers between 15 and 25. Draw and label a Venndiagram to represent the set P and indicate all the elements of set P in the Venn diagram. Solution: List out the elements of P. P = {16, 18, 20, 22, 24} ← „between‟ does not include 15 and 25 Draw a circle or oval. Label it P . Put the elements in P.

Example: Draw and label a Venn diagram to represent the set R = {Monday, Tuesday, Wednesday}. Solution: Draw a circle or oval. Label it R . Putthe elements in R.

Finite sets are sets that have a finite number of members. If the elements of a finite set are listed one after another, the process will eventually “run out” of elements to list. Example: A = {0, 2, 4, 6, 8, …, 100} C = {x : x is an integer, 1 < x < 10} An infinite set is a set which is not finite. It is not possible to explicitly list out all the elements of an infiniteset. Example: T = {x : x is a triangle} N is the set of natural numbers A is the set of fractions The number of elements in a finite set A is denoted by n(A). Example: If A is the set of positive integers less than 12 then A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} and n(A) = 11 If C is the set of numbers which are also multiples of 3 then C = {3, 6, 9, …} and C is an infinite set If Q is the set ofletters in the word ‘HELLO’ then Q = {H, E, L, O } , n(Q) = 4 ← „L‟ is not repeated.

Empty (NULL) Sets
There are some sets that do not contain any element at all. For example, the set of months with 32 days. We call a set with no elements the null or empty set. It is represented by the symbol { } or Ø . Some other example of null sets are: The set of dogs with six legs. The set of squares with 5sides. The set of cars with 20 doors. The set of integers which are both even and odd.

Set Equality
Consider the sets: P ={Tom, Dick, Harry, John} Q = {Dick, Harry, John, Tom} Since P and Q contain exactly the same number of members and the memebers are the same, we say that P is equal to Q, and we write P = Q. The order in which the members appear in the set is not important. Consider the...