The universal set is a set which contains elements of all the related sets, without repeating the  elements with itself. Let’s say if X and Y are two sets, such as X = {1,3} and Y =  {1,a,c}, then the universal set associated with these two sets is given by U = {1,3,a,c}.

Universal Set Definition

A set which contains all the elements of other sets, along with its own elements is known as the universal set. It is usually denoted by the symbol ‘U’.

• Let say , set P consists of all even numbers such that, P = {2, 4, 6, 8, 10, …} and set Q consists of all odd numbers, such that, Q = {1, 3, 5, 7, 9, …}. The universal set U consists of all natural numbers, such that, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….}. Therefore, as  very well known, all the even and odd numbers are a part of natural numbers. Therefore, Set U has all the elements of Set P and set Q , and is known as a universal set.

Complement of Universal Set 

We have a complement of sets for every set. The empty set is considered as the complement of the universal set. Which means where the Universal set consists of a set of all elements, the empty set contains no elements of the subsets. The empty set is also known as the Null set and is denoted by ‘{}’.

Making a universal set

If a universal set will be infinite, or even if it’s finite with a many number of elements, then the ellipses, the three dots (…), are used to represent the universal set. An example is given below :

U = {1, 2, 3, …}

Or,            U = {1, 2, 3, …, 4000}

Let’s solve some examples to develop a firm understanding of universal sets -:

Example 1 -: 

Consider, there are three sets, namely A , B and C . Given below are the elements of each set:

A={2, 4, 6}

B={3, 7, 8,}

C={4, 8, 10}

Find the universal set for all three sets A , B and C ?

Solution -:  As the universal set consists of all the elements without repetition of the given sets; thus, the universal set of A , B, and C is given by:

U={2, 3, 4, 6, 7, 8, 10}

Example 2 -: 

Let P is equal to a set of odd numbers, and Q  is equal to the set of even numbers. What will be the  universal set of P and Q ?

Solution -:  The sets P and Q can be written as:

P = {1, 3, 9…} and, 

Q = {2, 4, 6,…}

So, the universal set is:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10…}

From the above example, we observe that the universal set U of the sets P and Q results in the set of natural numbers.

Representation of Universal Set through Venn Diagram

• The Venn Diagram represents two or more circles used to demonstrate the relationships among different sets of finite elements or objects.  

• In the Venn diagram, the circles are used to demonstrate sets. These circles or sets are contained within the universal set, which is represented by a rectangle.

Difference Between Universal Set and Union of Set 

The universal set, including its elements, is a set that consists of all elements or objects. Whereas the union of sets is a two-set operation, say P and Q , resulting in a set with all the elements that belong to both sets P and Q.  

The following example will makes our understanding better towards the difference -: 

Example  -: 

Consider three sets with elements U = {3, 5}, set P = {a, s, d}, and set Q = {e, f, g}. Find the universal set U and the union of sets P and Q ?

Solution -: 

The universal set of the given three sets is given as follows:

U = { a, s, d, e, f, g, 3, 5}

Now, The union of set P and Q is given as:

   P U Q = {a, s, d, e, f, g}

Thus, the universal set is the combination of P and Q and U itself, whereas the union of P and Q contains Q and Q elements only.

Conclusion – :

The universal set is denoted by U and is a set that contains all the elements of all its subsets including its own elements, and the elements are not repeated. And on the other hand, the union of two sets is a set containing all elements that are present in A or in B (possibly both). We have talked about the cardinality of the universal set. We have even discussed the universal set and most importantly, the difference between the universal set and the union of sets.