Universal Set

In Mathematics, the collection of elements or group of objects is called a Set. There are various types of sets like Empty set, Finite set, Infinite set, Equivalent set, Subset, Superset and Universal set. All these sets have their own importance in Mathematics calculations. There is a lot of usage of sets in our day to day life but basically, they are used to represent bulk data or collection of data in a database. For example, our hand is a set of types of fingers. The notation of set is usually given by curly brackets, { } and each element in the set is separated by commas like {4, 7, 9}, where 4, 7, and 9 are the elements of sets.

Universal set contains a group of objects or elements which are available in all the sets and is represented in a Venn diagram. Suppose Set A consist of all even numbers such that, A = { 2, 4, 6, 8, 10, …} and set B consist of all odd numbers, such that, B = { 1, 3, 5, 7, 9, …}. The universal set U consist of all natural numbers, such that, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….}. Therefore, as we know, all the even and odd numbers are a part of natural numbers. Therefore, Set U has all the elements of Set A and Set B.

In this article, we are going to discuss Universal set, its usage and representation in a Venn Diagram with the help of examples.

Universal Set Definition

A universal set is a set which contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol ‘U’.(There is no standard notation for Universal set symbol, it can also be denoted by any other entity like ‘V’ or \(xi\)). Let us explain to you with the help of a universal set example.

Example: Let us say, there are three sets named as A, B and C. The elements of all sets A, B and C is defined as;

A={1,3,6,8}

B={2,3,4,5}

C={5,8,9}

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

Answer: By the definition we know, the universal set includes all the elements of the given sets. Therefore, Universal set for sets A,B and C will be,

U={1,2,3,4,5,6,8,9}

Explanation: From the above example, we can see that the elements of sets A, B and C are altogether available in Universal set ‘U’. Also, if you observe, no elements in the universal set are repeated and all the elements are unique.

Note: If Universal set contains Sets A, B and C, then these sets are also called subsets of Universal set. Denoted by;

\(A\subset U\) (A subset of U)

\(B\subset U\) (B subset of U)

\(C\subset U\) (C subset of U)

For Venn diagram representation of the universal set, we can take the example as;

U={heptagon} consisting of set A={pentagon, hexagon, octagon} and set C={nonagon}.

universal Venn diagram

We can understand the concept of Universal set also by taking an example of real world. Just like, U is a universal set. In this world, we have set of a human being, set of animals and also set of all living things, which we can consider as a subset of U. But we cannot consider a set of trees as subset of U.

There is a compliment of set for every set.The empty set is defined as the complement of the universal set. That means where Universal set consists of a set of all elements, the empty set contains no elements of the subsets. The empty set is also called a Null set and is denoted by \(\left \{ \right \}\).

Difference between Universal set and Union set

Student’s sometimes get confused with the universal set and union of the set, as both are same. But there is a difference between the two.

The universal set is a set which consists of all the elements or objects, including its own elements. But the union of two sets, say A and B, is a set which has all elements belonging either to set A and set B or both.

For example, Set A = {a,b,c} and set B={c,d,e} and U={1,2}. Therefore, the universal set for set A, B and U itself will be;

U ={a,b,c,d,e,1,2}

And if we have to write, the union of set A and B, then,

A∪B = {a,b,c,d,e}

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Practise This Question

A stadium is in circular shape. Within the stadium some areas have been allotted for a hockey court and a javelin range, as given in the figure. Assume the shape of the hockey court and the javelin range to be square and triangle, resp. The curators would like to accommodate a few more sports in the stadium. Help them by measuring the unallocated region within the stadium.(the radius of the stadium is 200 mts.)