Types 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