Sets and their representations

Sets are an organized collection of objects and can be represented in set-builder form or roster form. Usually, sets are represented in curly braces {}, for example, A = {1,2,3,4} is a set.  

Definition of Sets

Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set.

Elements of a Set

Let us take an example:

$A=\{1,2,3,4,5\}$

Set is represented by the capital letter.  Here, $A$ is the set and $\{1,2,3,4,5\}$ are the elements of the set or members of the set. The elements that are written in the set can be in any order but cannot be repeated. All the set elements are represented in small letter in case of alphabets.  Also, we can write it as $1\in A,2\in A$ ( $1$ belongs to $A$ , $2$ belongs to $A$ ) etc. The cardinal number of the set is 5. Some commonly used sets are as follows:

• $N:$ Set of all natural numbers
• $Z:$ Set of all integers
• $Q:$ Set of all rational numbers
• $R:$ Set of all real numbers
• $Z^{+}:$ Set of all positive integers

Order of Sets

The order of a set defines the number of elements a set is having. It describes the size of a set. The order of set is also known as the cardinality.

The size of set whether it is is a finite set or an infinite set, said to be set of finite order or infinite order, respectively.