What is Universal Set Mean in Math?

A universal set is a larger set formed from two or more sets, which are also considered subsets of the universal set. The universal set is represented by U and is a set containing all the members or objects of all its subsets, and the members are not repeated. 

Consider two sets, A = {1,2,3} and B = {1,a,b,c}, then the universal set associated with these two sets is U = {1,2,3,a,b,c}.

In Mathematics, the collection of well-defined distinct elements or groups of objects is called a Set.

There are different types of sets like Equivalent set, Empty set, Finite set, Infinite set, Subset, Superset, and Universal set. All these sets have their own importance in Mathematics. There is a lot of usage of sets in our day-to-day life, but normally they are used to denote a collection of data in a database.

The notation of a set is usually given by curly brackets, {}, and each member or object in the set is separated by commas like {5, 6, 9}, where 5, 6, and 9 are the members of sets.in this article learn about the universal set. Its definition is represented with solved examples.

What is a universal set?

A universal set is a collection of all elements or objects of all the related sets, known as its subsets. Or universal set is a collection of all members or elements of another set including its own element. it is usually represented by ‘U’.

Let the set of A consist of all odd numbers such that

A= {1,3,5,7,…} and set B consist of all even number such that B={2,4,6,8,…}  The universal set U consists 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 subset of natural numbers. Therefore, Set U has all the elements of Set A and Set B.

There is no formula to determine the universal set, we just have to denote all the members in a single which is collectively called a universal set. A universal set can be either a finite or infinite set.

Symbol of the universal set

The universal set is usually denoted by the symbol E or U. It consists of all the elements of its subsets, including its own elements.

For example:

Let’s considered, that there are three sets A, B, and C. The elements of all sets A, B, and C are defined as;

A={2,4,6}

B={1,3,7,9,11}

C={2,8,11}

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

Solution: according to the definition of a universal set, the universal set includes all the elements of the given sets.

U = {1, 2, 3, 4, 6, 7, 8, 9, 11}

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

Note: The sets A, B, and C are part of the universal set, then these sets are also called subsets of the Universal set.

• A ⊂ U (A is the subset of U)
• B ⊂ U (B is the subset of U)
• C ⊂ U (C is the subset of U)

Complement of the universal set

There is a complement of sets for every set. The Universal set consists of a set of all elements of all its related subsets, whereas the empty set contains no elements of the subsets. Thus, the complement of the universal set is an empty set, represented by ‘{}’ or the symbol ‘Φ’.

Summary

• If A is a subset of the universal set U, then all the members or objects in U that are not in A are called the complement of A.
• If A is a subset of the universal set U, then the complement of A is also a subset of U.
• The complement of a universal set is always an empty set.
• A set and its complement are the disjoint sets.