Universal Set: Definition, Symbol, Complement & Examples

Universal Set is a set that contains all related sets and subsets including itself. It is a set that includes the elements of all the related sets, without any repetition.

• Universal Set is denoted by the capital letter ‘U’.
• A set is defined as a well-organized collection of objects or items in Maths.
• The objects or items in a set are referred to as the ‘Elements’ of a set. 
• The elements in a Universal Set are not repeated, and thus, are unique. 

Example of Universal Set: Consider two sets A = {a, b, c} and B = {1, 2, 3, a, b}. Thus, the Universal Set will be U = {a, b, c, 1, 2, 3}.

Read More: NCERT Solutions for Class 11 Maths Sets

Key Terms: Universal Set, Sets, Union of Sets, Complement of Sets, Venn Diagram, Natural Numbers, Elements, Finite Set, Subsets

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What is a Set?

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Set is a well-defined collection of objects or elements. Sets are an integral part of the Set Theory proposed by Georg Cantor, a German mathematician.

• Sets are collections of elements like numbers, letters, etc.
• Elements of a set are the items or objects in a set.
• A set is denoted by a Capital Letter with its elements listed in Curly Brackets as A = {…..}.
• Sets are a crucial component of Mathematical Logic and organization. 

Sets can be classified into various types as follows: 

• Universal Set
• Empty Set
• Finite and Infinite Sets
• Equal Sets
• Subset
• Power Set
• Singleton Set
• Empty Set
• Disjoint Sets

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What is Universal Set?

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Universal Set is a set that consists of all the elements of all the related sets. There is no repetition of elements in a Universal Set.

• It is a collection of all elements or members of all the related sets, known as its subsets.
• Universal Set is denoted by the alphabet “U”. 
• Consider a set of all Natural Numbers as a Universal Set N.
• Sets of even and odd numbers or prime numbers are subsets of Universal Set N. 

Universal Set Definition

Universal Set is defined as a set that contains all the elements or objects of other sets, including its own elements. It is usually denoted by the letter U or E. It can be either a finite or infinite set.

Consider that a Universal Set U consists of Sets A, B, and C. These sets are referred to as Subsets of Universal Set U and are denoted as

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

Symbol of Universal Set

• Universal Set is denoted by the symbol U or E.
• It is composed of all the elements of its subsets, including its own elements.

Sets Detailed Video Explanation

Complement of Universal Set

There is a complement for every set in Set Theory. An Empty Set is considered to be a complement of a Universal Set.

• Universal set is a set of all elements of all its related subsets.
• Empty set has no elements of the subsets.
• Therefore, an empty set is the complement of a universal set.
• Empty set is also referred to as a Null set and is denoted by ‘{}’ or the symbol 'Φ'.

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Universal Set Example

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Universal Set is defined as a larger set that contains elements of all the related sets, without any repetition of elements. Consider a Universal Set N which is a set of Natural Numbers. Now, consider the given sets: 

• Set of Even Numbers, A = {2, 4, 6, 8, 10, …}
• Set of Odd Numbers, B = {1, 3, 5, 7, 9, …}

So, the Universal Set U consists of all natural numbers which include both even and odd numbers, such that, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….}. Thus, Set U is a universal set that has all the elements of Set A and Set B.

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Venn Diagram of Universal Set

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Venn Diagram are used to show the relationship between sets in form of pictorial representation.

• They are basically the graphical representation of Sets.
• Universal Set is represented by a rectangle in Venn Diagrams.
• The subsets are represented by circles or ovals. 

Consider a Universal Set U along with its two subsets A and B. 

• Universal Set U = {1, 2, 3, 4, 5, 6, 7, 8}.
• Set A = {2, 3, 4}
• Set B = {6, 7, 8}

Thus, the Venn Diagram of the Universal Set will be as follows: 

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Universal Set and Union of Sets

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There is a persisting confusion between the Union of Sets and the Universal Set. 

• Universal Set consists of all the elements or objects, including its own elements.
• Union of Sets is a set operation that results in a set that has all elements belonging to either of the given sets. 
• Universal Set is denoted by ‘U’, while Union of Sets is denoted by ‘∪’.

Example: Consider a Universal Set U and two other sets A and B.

• U = {Dog, Cow, Lion}
• A = {1,6,9}
• B = {k,l,h,g}

Universal Set of the given sets will be 

U = {1, 6, 9, k, l, h, g, Dog, Cow, Lion}

It includes all the elements of Sets A, B, and the universal set itself. 

Union of Sets will be given as

A U B = {1, 6, 9, k, l, g, h}

Union of sets includes the elements of only Sets A and B.

Read More: ​Sets Important Questions

Difference Between Universal Set and Union of Sets

The difference between Universal Set and Union of Sets is as follows: 

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Things to Remember

• Universal Set is a type of set that includes all the elements of other sets, including its own elements.
• It is a larger set that contains elements of all the related sets, without any repetition.
• Universal Sets are dented by ‘U’ or ‘E’ in Set Theory.
• An Empty Set is the complement of a Universal Set.
• Union of Sets is different from Universal Set as it includes all elements belonging to either set or both.
• Sets are defined as well-organized collections of elements in Maths. 

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Previous Years’ Questions

1. Two sets A and B are as under: A = {(a,b)... (JEE Main - 2018)
2. Let A, B, C be finite sets. Suppose that n(A) = 10, n(B) = 15… (BITSAT - 2017)
3. If A and B are not disjoint sets, then n(A∪B)...
4. If A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B)... (VITEEE - 2018)
5. Two finite sets A and B have m and n elements respectively. If… (KEAM)
6. The set of all real numbers x for which… (JEE Advanced - 2002)
7. Two finite sets have m and n elements. The total number of subsets…
8. Two finite sets have m and n elements. The total number…
9. In a certain town 25% families own a cell phone, 15% families own… (KEAM)
10. In a class of 140 students numbered 1 to 140, all even-numbered students… (JEE Main - 2019)
11. If A and B are non-empty sets such that… (KEAM)
12. A survey shows that 63% of Americans like cheese where as… (JKCET - 2013)
13. If a class of 175 students the following data shows the number… (COMEDK UGET - 2015)
14. If a set A has 4 elements, then the total number of proper subsets of set… (COMEDK UGET - 2015)