# Sets

An introduction of sets and its definition in mathematics. The concept of sets is used for the foundation of various topics in mathematics.

To learn sets we often talk about the collection of objects, such as a set of vowels, set of negative numbers, a group of friends, a list of fruits, a bunch of keys, etc.

What is set (in mathematics)?

The collection of well-defined distinct objects is known as a set. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not. The word ‘distinct’ means that the objects of a set must be all different.

For example:

1. The collection of children in class VII whose weight exceeds 35 kg represents a set.

2. The collection of all the intelligent children in class VII does not represent a set because the word intelligent is vague. What may appear intelligent to one person may not appear the same to another person.

Elements of Set:

The different objects that form a set are called the elements of a set. The elements of the set are written in any order and are not repeated. Elements are denoted by small letters.

Notation of a Set:

A set is usually denoted by capital letters and elements are denoted by small letters

If x is an element of set A, then we say x ϵ A. [x belongs to A]

If x is not an element of set A, then we say x ∉ A. [x does not belong to A]

For example:

The collection of vowels in the English alphabet.

Solution :

Let us denote the set by V, then the elements of the set are a, e, i, o, u or we can say, V = [a, e, i, o, u].

We say a ∈ V, e ∈ V, i ∈ V, o ∈ V and u ∈ V.

Also, we can say b ∉ V, c ∉ v, d ∉ v, etc.

Set Theory

Sets

Objects Form a Set

Elements of a Set

Properties of Sets

Representation of a Set

Different Notations in Sets

Standard Sets of Numbers

Types of Sets

Pairs of Sets

Subset

Subsets of a Given Set

Operations on Sets

Union of Sets

Intersection of Sets

Difference of two Sets

Complement of a Set

Cardinal number of a set

Cardinal Properties of Sets

Venn Diagrams