Sets, in mathematics, 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. Also, check the set symbols here.

In sets theory, you will learn about sets and it’s properties. It was developed to describe the collection of objects. You have already learned about the classification of sets here. The set theory defines the different types of sets, symbols and operations performed.

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.

What are the Elements of a Set

Let us take an example:

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

Since a set is usually represented by the capital letter.  Thus, 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 ∈ A, 2 ∈ 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.

Also, check:

Representation of Sets

The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either the Statement form, Roster Form or Set Builder Form.

Statement Form

In statement form, the well-defined descriptions of a member of a set are written and enclosed in the curly brackets.

For example, the set of even numbers less than 15.

In statement form, it can be written as {even numbers less than 15}.

Roster Form

In Roster form, all the elements of a set are listed.

For example, the set of natural numbers less than 5.

Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….

Natural Number less than 5 = 1, 2, 3, 4

Therefore, the set is N = { 1, 2, 3, 4 }

Set Builder Form

The general form is, A = { x : property }

Example: Write the following sets in set builder form: A={2, 4, 6, 8}

Solution:

2 = 2 x 1

4 = 2 x 2

6 = 2 x 3

8 = 2 x 4

So, the set builder form is A = {x: x=2n, n ∈ N and 1  ≤ n ≤ 4}

Also, Venn Diagrams are the simple and best way for visualized representation of sets.

Types of Sets

We have several types of sets in Maths. They are empty set, finite and infinite sets, proper set, equal sets, etc. Let us go through the classification of sets here.

Empty Set 

A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø.

A set of apples in the basket of grapes is an example of an empty set because in a grapes basket there are no apples present.

Singleton Set 

A set which contains a single element is called a singleton set.

Example: There is only one apple in a basket of grapes.

Finite set

A set which consists of a definite number of elements is called a finite set.

Example: A set of natural numbers up to 10.

A = {1,2,3,4,5,6,7,8,9,10}

Infinite set 

A set which is not finite is called an infinite set.

Example: A set of all natural numbers.

A = {1,2,3,4,5,6,7,8,9……}

Equivalent set

If the number of elements is the same for two different sets, then they are called equivalent sets. The order of sets does not matter here. It is represented as:

 n(A) = n(B)

where A and B are two different sets with the same number of elements.

Example: If A = {1,2,3,4} and B = {Red, Blue, Green, Black}

In set A, there are four elements and in set B also there are four elements. Therefore, set A and set B are equivalent.

Equal sets 

The two sets A and B are said to be equal if they have exactly the same elements, the order of elements do not matter.

Example: A = {1,2,3,4} and B = {4,3,2,1}

A = B

Disjoint Sets 

The two sets A and B are said to be disjoint if the set does not contain any common element.

Example: Set A = {1,2,3,4} and set B = {5,6,7,8} are disjoint sets, because there is no common element between them.

Subsets

A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is considered to be the subset of another set. In general, a subset is a part of another set.

Example: A = {1,2,3}

Then {1,2} ⊆ A.

Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.

Note: The set is also a subset of itself.

If A is not a subset of B, then it is denoted as A⊄B.

Proper Subset

If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B.

Example: If A = {2,5,7} is  a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7}

But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.

Superset

Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is represented as A ⊃ B.

For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B.

Universal Set

A set which contains all the sets relevant to a certain condition is called the universal set. It is the set of all possible values. 

Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:

U = {1,2,3,4,5}

Operations on Sets

In set theory, the operations of the sets are carried when two or more sets combine to form a single set under some of the given conditions. The basic operations on sets are:

• Union of sets
• Intersection of sets
• A complement of a set
• Cartesian product of sets.
• Set difference

Basically, we work more on union and intersection of sets operations, using Venn diagrams.

Union of Sets

If set A and set B are two sets, then A union B is the set that contains all the elements of set A and set B. It is denoted as A ∪ B.

Example: Set A = {1,2,3} and B = {4,5,6}, then A union B is:

A ∪ B = {1,2,3,4,5,6}

Intersection of Sets

If set A and set B are two sets, then A intersection B is the set that contains only the common elements between set A and set B. It is denoted as A ∩ B.

Example: Set A = {1,2,3} and B = {4,5,6}, then A intersection B is:

A ∩ B = { } or Ø

Since A and B do not have any elements in common, so their intersection will give null set.

Complement of Sets

The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by P’.

Properties of Complement sets

1. P ∪ P′ = U
2. P ∩ P′ = Φ
3. Law of double complement : (P′ )′ = P
4. Laws of empty/null set(Φ) and universal set(U),  Φ′ = U and U′ = Φ.

Cartesian Product of sets

If set A and set B are two sets then the cartesian product of set A and set B is a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. It is denoted by A × B.

We can represent it in set-builder form, such as:

A × B = {(a, b) : a ∈ A and b ∈ B}

Example: set A = {1,2,3} and set B = {Bat, Ball}, then;

A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}

Difference of Sets

If set A and set B are two sets, then set A difference set B is a set which has elements of A but no elements of B. It is denoted as A – B.

Example: A = {1,2,3} and B = {2,3,4}

A – B = {1}

Sets Formulas

Some of the most important set formulas are:

Also, check the set symbols here.

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