You will learn:
In mathematics, a Set is a well-defined collection of distinct objects or, in other words, it is just a group of things with a certain property in common. For example numbers 1, 3, 6, 10 are distinct objects when considered separately, but when they are considered collectively they form a single set of size 4, written as {1,3,6,10}. Few more examples:
Objects used to form a set are called elements or members of a set. A set is defined by describing the contents or listing the elements of the set in curly brackets where each element is separated by a comma(,).
If A is a set of colors: Green, Blue, Yellow and Red then
set A = {Green, Blue, Yellow, Red}
- We use a capital letter to represent a set (here it is denoted as A).
- Elements of set A are Green, Blue, Yellow and Red.
- Color Green 'belongs' to set A, this is denoted as \(\textrm{Green} \in A\).
- Color Black 'does not belong' to set A, this is denoted as \(\textrm{Black} \notin A\).
- The order of elements in the set is not important. We can write A = {Blue, Yellow, Green, Red}
A set that contains no elements, { } is called an Empty set and is denoted as ΓΈ.
Let's take another set B = {Yellow, Green, Red}. Notice that B has all the colors that are elements of set A. Therefore we say B as a subset of A and we write as \(B \subset A\).
Representation of a Set
A set can be represented by various methods. 3 common methods used are:
Let's take an example and define the set according to these three forms:
Statement form: Well defined description of the elements of the set is given. Example: Set of natural numbers less than 6
Roster form: Elements are listed within pair of brackets {} and separated by commas. Above Example in Roaster form is: N = {1, 2, 3, 4, 5 }
Set Builder form: Set is described by a property that its member must satisfy. N = { x : x is natural number less than 6}
Equal sets: Two sets are said to be equal if both have the same elements. For example A = {1, 3, 4, 6} and B = {3, 4, 1, 6} are Equal sets.
Size of a set: Size of a set is known as Cardinality number, denoted by |A| ( A is a set). Example: A = {Blue, Yellow, Green, Red}, Cardinality of set A is 4, i.e.
\(|A| = 4\)
The size of a set can be finite or infinite. A set having a finite number of elements is said to be a Finite set. Like { 1, 2, 3, 4, 5} is a finite set whose cardinality is 5. The set having uncountable elements is the Infinite set. For example, a set of all integers is an infinite set. The infinite set has little different representation than the finite set. For example: Set of all whole numbers is an infinite set and is represented as : W = {1, 2, 3, 4, ... } Here three dots means 'goes on forever'.
Symbols used for number types:
Natural numbers: N, Whole Numbers: W, Integers: Z, Rational numbers: Q, Real numbers: R,
