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## What are sets in mathematics?

A set in mathematics is simply an organized collection of well-defined objects or elements. In other words, a set refers to the collection of elements that are unique. It can have any group of collected items such as type of automobiles, days of a week, letter of alphabets, etc. The representation of a set is done by a capital alphabetic letter.

__Elements of a set__:

Assume a set A = {7,8,9}. In this A is the set and 7, 8, 9 are the elements or members of a set. The elements in a set can’t be repeated but can be in any sequence and are enclosed in a curly bracket. When an element is a part of a set then it is denoted by ‘∈’ and when an element is not part of a set then it is denoted by ‘∉’.

__The cardinality of a set__:

The cardinal number, order of a set, or cardinality of a set implies a total number of elements in a set. The sets are a collection of unique elements and all elements in a set are related to each other.

## How a set can be represented?

__Semantic form__: This form is used to show what are the elements of a set. For Example, Set B has the first four even natural numbers.__Roster Form__: Most common way to represent the set is curly brackets {}. Here elements are separated by commas and their order doesn’t matter. For Example, Set A = {4,5,6,7}.__Set Builder Form__: In this form, a vertical bar is used in its representation, with a text that describes the character of elements of a set. For example, B = {k| k is an odd number, k ≤ 10}.

## What are the set formulas?

### Following are the most important set formulas:

For any two given sets, say A and B,

- n(A U B) = n(A) + n(B) – n(A ∩ B)
- n (A ∩ B) = n(A) + n(B) – n(A U B)
- n(A) = n(A U B) + n(A ∩ B) – n(B)
- n(B) = n(A U B) + n(A ∩ B) – n(A)
- n(A – B) = n(A U B) – n(B)
- n(A – B) = n(A) – n(A ∩ B)

## For any two disjointed sets A and B,

- n(A U B) = n(A) + n(B)
- A ∩ B = ∅
- n(A – B)= n(A)
__Operations on a set__:__Union of sets__: It is denoted as A ∪ B, a set that contains all elements of two sets set A and set B. For example, Set A = {1,2,3} and Set B = {4,5,6}, then A ∪ B = {1,2,3,4,5,6}.__Intersection of sets__: It is denoted as A ∩ B, a set that contains all common elements of two sets set A and set B. For example, Set A = {1,2,3} and Set B = {3,4,5}, then A ∩ B = {3}.__Complements of sets__: The complement of any set, say B, is the set of all elements that are not in set B. It is represented by B’.__Cartesian Product of sets__: Cartesian product of sets, say set A and set B is a set of all ordered pairs (a,b), where, element of set A is a and, the element of set B is b. For example, set A = {1,2} and set B = {chair, table}, then A × B = {(1×chair),(1×table),(2×chair)(2×table)}.__Difference of sets__: Set A difference Set B is a set that doesn’t have elements of B but A. For example, A = {1,2,3} and B = {2,3,4}, then A – B = {1}.

There are different types of sets including several formulas and operations. For a better understanding, we have discussed the basics of the set in this article. Facing difficulty in understanding what set is? Just go to the Cuemath website or read the below-mentioned article.