Posted by Anjali Kaur on Jun 03, 2020

What Are The Different Types Of Sets?

So far we have discussed the meaning of Sets and How we write the elements of a set? Now, we will see different types of sets:

The following are the types of sets:

  1. Subset
  2. Proper and Improper subset
  3. Finite Set
  4. Infinite Set
  5. Null Set
  6. Singleton Set
  7. Equal and Equivalent Set
  8. Universal Set
  • Subset: It is the part of a set. For example, if we have two sets A and B as:

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

B = {2,3,5}

As you can see each element of B is contained in A (every number 2,3,5 of B set is already in A set with some extra numbers), we call B as a subset of A. Similarly, we also call A as a super set of B. Check this image:

  • Proper and Improper Subset: Proper subset is that subset where the subset is not equal to super set but it is contained in super set and improper subset is that where the elements of each set are equal to each other. I will explain with an example, consider set A,B and C as follows:

A = {1,2,4,7}

B = {2,4,7}

C = {4,1,2,7}

Now, first focus on set A and B. You can see that each element of B is already in set A with some other element also. So, B is smaller than A, hence B is termed as a ‘proper subset’ of A.

Now, look at set A and C. each element of C is already in A (sequence of elements do not matter in set theory), you can also rephrase it and say each element of A is already in C. Both statements are correct. Therefore, we say that ‘C is an improper subset of A’ or ‘A is an improper subset of C’. Check the below image for the symbols we use for proper and improper subset:

  • Finite Set: Those sets whose elements are countable. For example,

A = {1,2,4,6,10} here we have 5 elements, this is an finite set. Let’s take a look at another example,

B = {1,2,3,……….., 1000000000000000000000000000000000000000000000000}

Well, I know it will be difficult to calculate, but yes it is also a countable set and finite set. So, under finite set; upper limit and lower limit should be given.

  • Infinite Set: Those sets whose elements are uncountable. Like, if

A = {1,2,3,4,………….} then it is an infinite set, as we don’t know the upper limit of this set.

  • Null Set/ Void Set/ Empty Set: Those sets which contains nothing, literally nothing, not even zero are called empty sets. Note: Null set is always a subset of every set. Now, let’s take a look at the example of null set with the help of an image:
  • Singleton Set: A set which contains only one element in it are called singleton set. For example, {1}, {0}, {100000000}, etc are all singleton sets.
  • Equal and Equivalent Set: Equal sets are the improper subsets, i.e., when each element of a set matches with the other set, we call them equal set. Equivalent sets are those when just the number of elements in each sets are same, it does not matter whether there elements matches or not. Let’s understand it better with the help of an example as shown below:

As, you can see in Set A elements are 5,6,7, 8 and in Set D elements are 6,7,7,8. Elements are same in each sets, so we termed A and D as equal sets.

  • Universal Sets:These are like universe which contains everything, they are the maker of every set. So, if we have three sets A, B, U as:

A = {1,2}

B = {5,6}

U = {1,2,3,4,5,6}

So, for set A and set B, U is the universal set, as U contains each element of set A and B.

If you have any doubt regarding the types of sets then drop an email: contact@LearnWithAnjali.com

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