In Mathematics, sets are defined as the collection of objects whose elements are fixed and can not be changed. In other words, a set is well defined as the collection of data that does not carry from person to person. The elements can not be repeated in the set but can be written in any order. The set is represented by capital letters.

The Empty set, finite set, equivalent set, subset, universal set, superset, infinite set are some types of set. Each type of set has its own importance during calculations. Basically, in our day to day life, sets are used to represent bulk data and collection of data. So, here in this article, we are going to learn and discuss the Universal Set.

What is Set, What are Types of Sets and Their Symbols?

A set is well defined as the collection of data that does not carry from person to person.

1. Empty Sets -

The set, which has no elements, is also called a Null set or Void set. It is denoted by {}.

Below are the two example of empty set.

Example of empty set: Let, Set A = {a : a is the number of students studying in Class 6th and Class 7th}. Since we all know, a student cannot learn in two classes, therefore set A is an empty set.

Another example of empty set is, set B = {a: 1 < a < 2, a is a natural number}, we know natural cannot be a decimal, therefore set B is a null set or empty set.

2. Singleton Sets-

The set which has just one element is named a singleton set.

For Example: Set A = { 8 } is a singleton set.

3. Finite and Infinite Sets-

A set which has a finite number of elements is known as finite sets, whereas the set whose elements can't be estimated, but it has some figure or number, which is large to precise in a set which is known as Infinite Set.

For Example: Set A = {3,4,5,6,7} is a finite set, as it has a finite number of elements.

Set C = {Number of Cows in India} is an infinite set, there is an approximate number of Cows in India, but the actual number of cows cannot be expressed, as the numbers could be very large and counting all cows is not possible.

4. Equal Sets-

If every element of set A is also the elements of set B and if every element of set A is also the elements of set A are called equal sets. It means set A and set B have equivalent elements and that we can denote it as:

A = B

For Example: Let A = {3,4,5,6} and B = {6,5,4,3}, then A = B

And if A = {set of even numbers} and B = { set of natural numbers} the A ≠ B, because natural numbers consist of all the positive integers starting from 1,2,3,4,5 to infinity, but even numbers start with 2,4,6,8, and so on.

5. Subsets-

A set S is said to be a subset of set T if the elements of set S belong to set T, or you can say each element of set S is present in set T. Subset of a set is denoted by the symbol (⊂) and written as S ⊂ T.

We can also write the subset notation as;

S ⊂ T if p ∊ S ⇒ p ∊ T

According to the equation given above, “S is a subset of T only if “p” is an element of S as well as an element of T.”Each set is a subset of its own set, and a void set or empty set is a subset of all sets.

6. Power Sets-

The set of all subsets is known as power sets.We know the empty set is a subset of all sets, and each set is a subset of itself. Taking an example of set X = {2,3}. From the above-given statements, we can write,

{} is a subset of {2,3}

{2} is a subset of {2,3}

{3} is a subset of {2,3}

{2,3} is also a subset of {2,3}

Therefore, power set of X = {2,3},

P(X) = {{},{2},{3},{2,3}}

7. Universal Sets-

A set that contains all the elements of other sets is called universal sets. Generally, it is represented as ‘U.’

For Example: set A = {1,2,3}, set B = {3,4,5,6} and C = {5,6,7,8,9}

Then, we will write universal set as, U = {1,2,3,4,5,6,7,8,9,}

Note: According to the definition of the universal set, we can say that all the sets are subsets of the universal set.

Therefore,

A ⊂ U

B ⊂ U

And C ⊂ U

8. Disjoint Sets

If two sets X and Y do not have any common elements, and their intersection results in zero(0), then set X and Y are called disjoint sets.It can be represented as; X ∩ Y = 0.

Union, Intersection,Difference and Complement of Sets-

1. Union of Sets-

The union of two sets consists of all their elements. It is denoted by (⋃).

For Example: Set A = {2,3,7} and set B = { 4,5,8}

Then the union of set A and set B will be;

A ⋃ B = {2,3,7,4,5,8}

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2. Intersection of Sets-

The set of all elements, which are common to all the given sets, gives an intersection of sets. It is denoted by ⋂.

For Example: set A = {2,3,7} and set B = {2,4,9}

So, A ⋂ B = {2}

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3. Difference of Sets-

The difference between set S and set T is such that it has only those elements which are in the set S and not in the set T. S – T = {p : p ∊ S and p ∉ T}

Similarly, T – S = {p: p ∊ T and p ∉ S}

4. Complement of a Set

Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U - A),is defined as

A’ = {x U : x A}

Clearly, x A’ x A

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Every set has a complement of sets. Also, for a universal set, the empty set is known as the complement of the universal set. The empty set contains no elements of the subset and is also known as Null Set, which is denoted by {Ø} or {}.

Questions to be Solved-

Question 1) If set A = {a, b, c, d} and B = {b, c, e, f} then, find A-B.

Answer) Let’s find the difference of the two sets,

A – B = {a, d} and B – A = {e, f}

Question 2) Let X = {David, Jhon, Misha} be the set of students of Class XI, who are in the school hockey team. Let Y = {Zoya, Rahul, Riya} be the set of students from Class XI who are in the school football team. Find X U Y and interpret the set.

Solution: (U Union - Combination of Two Sets)

Given X = {David, Jhon, Zoya}

Y = {Zoya, Rahul, Riya}

Common elements (Zoya) should be taken once

X U Y = {David, Jhon, Zoya, Rahul, Riya}.

This union set is equal to the set of students from Class eleven who are present in the hockey team or in the football team or in both of the teams.

FAQ (Frequently Asked Questions)

Question 1. What is Set, What are the Different Types of Sets and their Symbols?

Answer. Following are the Different Types of Sets in Set Theory:

Empty set.

Singleton set.

Finite and Infinite set.

Union of sets.

Intersection of sets.

Difference of sets.

Subset of a set.

Disjoint sets

Question 2. What is a Venn Diagram?

Answer. A Venn diagram is a pictorial representation of different sets. It helps the student to visualize the logical relationship between different sets. Venn diagrams make it easier to understand the difference and similarities between the sets. Venn diagrams are generally elements of the sets represented in a circle. And overlapping of circles represents the elements common in the sets.

Example: Consider two sets A and B. Set A is the collection of numbers which are multiples of 5 and Set B is the collection of the even numbers.

Hence when the Venn diagram is drawn the number 20, 10, and 30 are common in both the sets, therefore, represented in overlapping circles.