Types of Sets

In the field of Mathematics, sets can be defined as the collection of objects whose elements are fixed and cannot be changed. You can say that a set is a well defined collection of objects. The elements cannot be repeated in a set but can be written in any order. The set is always represented by capital letters.

What are the types of Sets?

There are primarily 8 types of sets that are used in Mathematics, they are:

1. Empty Sets - The set, which has no elements, is also called a Null set or Void set. It is denoted by {}.

2. Singleton Sets - The set which has just one element is named a singleton set.

3. Finite and Infinite Sets - A set that has a finite number of elements is known as a finite set, whereas the infinite set is the set whose elements can't be estimated, but it has some figure or number that is adequate enough to evaluate that set.

4. Equal Sets - If every element of set A is also the element of set B and if every element of set A is also the elements of set A are called equal sets. This implies that the elements of both the sets i.e. set A and set B are equal.

5. Subsets - A set P is said to be a subset of set U if the elements of set U belong to set P. In other words, it can be said that each and every element present in the  set P is also present in set U. 

6. Power Sets - The set of all subsets is known as power sets. 

7. Universal Sets - A set that contains all the elements of other sets is called a universal set. 

8. Disjoint Sets - If two sets X and Y do not have any common elements, and their intersection results in zero (0), then set X and Y are called disjoint sets.

Union, Intersection,Difference and Complement of Sets -

Union of Sets -

The union of two sets consists of all their elements. It is denoted by (⋃).

For example: Set A = {2,3,7} and set B = { 4,5,8}

Then the union of set A and set B will be;

 B ⋃ B = {2,3,7,4,5,8}

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Intersection of Sets -

The set of all elements, which are common to all the given sets, gives an intersection of sets. It is denoted by ⋂.

For Example: set A = {2,3,7} and set B = {2,4,9}

So, A ⋂ B = {2}

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Difference of Sets -

The difference between set S and set T is such that it has only those elements which are in the set S and not in the set T. S – T = {p : p ∊ S and p ∉ T}

Similarly, T – S = {p: p ∊ T and p ∉ S}

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Complement of a Set

Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U - A),is defined as:

A’ = {x   U : x  A}

Clearly, x  A’  x  A

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Every set has a complement of sets. Also, for a universal set, the empty set is known as the complement of the universal set. The empty set contains no elements of the subset and is also known as Null Set, which is denoted by {Ø} or {}.

Questions to be Solved:

Solved Examples

1.If set A = {a, b, c, d} and B = {b, c, e, f} then, find A-B.

Answer: Let’s find the difference of the two sets,

A – B = {a, d} and B – A = {e, f}

2.Let X = {David, Jhon, Misha} be the set of students of Class XI, who are in the school hockey team. Let Y = {Zoya, Rahul, Riya} be the set of students from Class XI who are in the school football team. Find X U Y and interpret the set.

Solution:  

(U Union - Combination of two sets)

Given X = {David, Jhon, Zoya}

 Y = {Zoya, Rahul, Riya}

Common elements (Zoya) should be taken once

X U Y = {David, Jhon, Zoya, Rahul, Riya}.

This union set is equal to the set of students from Class eleven who are present in the hockey team or in the football team or in both of the teams.