In this chapter we will learn the concept of universal set with examples and its properties.
In the end we will also learn universal set representation through Venn Diagram.
To understand the chapter you should have basic understanding of the concept of sets, subsets and their properties.
Universal Set definition
Definition 01
Universal set is collection of all the possible elements related to particular topic.
For example, if you are considering name of countries in the world then Universal set will contain the name of all 195 countries in the set.
Some other examples of universal sets are:
( i ) U = { Name of all the Oceans }
( ii ) U = { Name of all mammals living in earth }
( iii ) U = { All Natural Numbers }
Hence, if you are considering any particular topic, the universal set will contain all the possible elements related to that topic.
This definition of universal set is valid for general topic.
The below definition if useful when two or more set is given and you have to find the universal set for the given scenario.
Definition 02
For given two or more sets, the universal set is the collection of all the unique elements present in the given set.
For Example, let two sets A & B are given :
A = { 5, 10, 15 }
B = { 15, 17 }
Considering the above sets A & B, the universal set will be:
U = { 5, 10, 15, 17 }
Note that the universal set contain all the unique elements of both set A and set B.
Types of Universal Set
Universal set can be finite or infinite set.
If the universal set contain countable number of elements than it is termed as Finite Universal Set.
For Example;
U = { 2, 4, 6, 8, 10, 12 }
U = { Asia, North America, South America, Europe, Africa, Australia, Antarctica}
If the universal set contain unending number of elements than it is termed as Infinite Universal set.
For Example
U = { Set of natural numbers }
U = { Collection of prime numbers }
Representing Universal Set
The universal set is represented by letter ” U “
Hence, whenever you are asked to represent a universal set, you have to name it “U” for our convinience.
Representing Universal Set in Venn Diagram
In Venn diagram, the sets & its elements is represented by circle.
The universal set is represented by rectangular box inside which circles are drawn.
Let us understand the concept with example.
Consider two sets A & B with following elements.
A = { 2, 4, 6, 8, 10 }
B= { 7, 11 }
The universal set is given as:
U = { Collection of natural numbers }
U = { 1, 2, 3, 4, 5 . . . . }
For set A, B and U, the Venn diagram is shown as:
In the above diagram;
The universal set U is shown by the rectangle which contains all the natural numbers.
Set A is shown by green circle.
It is inside the rectangle box because the elements of set A is present inside the universal set U.
Similarly set B is shown by blue circle inside the rectangle box.
Conclusion
In Venn diagram, the universal set is shown by a rectangular box inside which all the sets are drawn using circle.
Solved Problems on Universal Set
(01) Consider the below sets;
A = { Artic Ocean, Atlantic Ocean }
B = {Pacific Ocean}
If these are the above sets then what will be the universal set?
Solution
The above sets depicts ocean’s name.
The universal set then contain list of all oceans in the world.
U = { List of all oceans in the world }
(02) Given below are two sets A & B
A = { 5, 11, 19, 31}
B = { 18, 19, 20, 21}
Find the universal set for the given set A & B
Solution
The Universal set can be found by writing down unique elements of all the given sets.
U = { 5, 11, 18, 19, 20, 21, 31}
(03) Given below are three sets X, Y & Z
X = { 45, 49 }
Y = { 41, 42, 43 }
Z = { 45, 46, 47, 48 }
Find the universal set for the given set X, Y & Z
Solution
For universal set, write down all the unique elements present in the set X, Y & Z.
Hence, the universal set is:
U = { 41, 42, 43, 4, 46, 47, 48, 49 }
(04) Given below is the Universal set U.
U = { 2, 4, 6, 8, 10, 12, 14, 16 . . . .}
Identify if the below set is subset of U or not?
(i) A = { 1, 2, 4, 6 }
(ii) B = { 24, 36, 58, 48 }
(iii) C = { 100, 124, 136 }
Solution
The universal set given in question is:
U = { 2, 4, 6, 8, 10, 12, 14, 16 . . . .}
The given universal set is actually collection of positive even numbers.
It is an infinite set as the elements are never ending.
(i) A = { 1, 2, 4, 6 }
Set A is not a subset of U because it contains odd number 1 which is not found in universal set.
(ii) B = { 24, 36, 58, 48 }
Set B is a subset of U.
All the elements of Set B are even numbers which are also present in given universal set U.
(iii) C = { 100, 124, 136 }
Set C is also subset of U.
All the elements of set C are even numbers which are also present in U.
(05) Three sets are given below. Identify the universal set among the three.
A = { x : x = 2n+1 & x 𝜖 N }
B = { x : x = 2n & x 𝜖 N }
C = { x : x = n & x 𝜖 N }
Solution
Let us observe each of the below set step by step.
A = { x : x = 2n+1 & x 𝜖 N }
Set name A.
Set A is collection of elements x such that;
⟹ x = 2n + 1
⟹ x belongs to natural numbers
Set A can be written in roster form as:
A = { 3, 5, 7, 9, 11 . . . . }
Similarly Set B & C can be written as following;
B = {2, 4, 6, 8, 10 . . . . }
C = { 1, 2, 3, 4, 5, 6 . . . .}
Observe that
Set A = { Collection of positive odd numbers from 3 }
Set B = { Collection of +ve even numbers }
Set C = { collection of natural number }
Clearly set C is the universal set as it contains all the elements of Set A & Set B.