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What is Set, Types of Sets and Their Symbols?

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Sets in Maths

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, and 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 examples of the empty set.


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


Another example of an empty set is  set B = {a: 1 < a < 2, a is a natural number}, we know a natural number 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 that has a finite number of elements is known as a finite set, whereas the set whose elements can't be estimated, but has some figure or number, which is large to precise in a set, 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 B is also the elements of set A, then sets A and B 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} then 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 a universal set. 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 the 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}

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.

Answer:  (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 XI who are present in the hockey team or the football team or both of the teams.

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FAQs on What is Set, Types of Sets and Their Symbols?

1. What is a set in mathematics as per the CBSE Class 11 syllabus?

In mathematics, a set is a well-defined collection of distinct objects, known as elements or members. For a collection to be a set, the criteria for including an object must be clear and unambiguous. Sets are typically denoted by capital letters (e.g., A, B, C) and their elements are listed within curly braces {}. For example, the set V of vowels in the English alphabet is written as V = {a, e, i, o, u}.

2. What are the main types of sets students learn about?

The main types of sets studied in mathematics include:

  • Empty Set (or Null Set): A set containing no elements, denoted by {} or ∅.
  • Singleton Set: A set that has only one element.
  • Finite Set: A set with a limited, countable number of elements.
  • Infinite Set: A set with an unlimited number of elements.
  • Equal Sets: Two sets that have the exact same elements.
  • Subset: A set A is a subset of set B if all elements of A are also present in B.
  • Power Set: The set of all possible subsets of a given set.
  • Universal Set: A set containing all elements relevant to a particular context, denoted by U.

3. What are the common symbols used in set theory and what do they mean?

Here are the most common symbols used when working with sets:

  • {} or ∅: Represents the Empty Set (a set with no elements).
  • ∈: Means "is an element of." For example, 3 ∈ {1, 2, 3}.
  • ∉: Means "is not an element of."
  • ⊂: Represents a proper subset. A ⊂ B means A is a subset of B, but A ≠ B.
  • ⊆: Represents a subset. A ⊆ B means all elements of A are in B.
  • ∪: Represents the Union of sets (all elements from both sets combined).
  • ∩: Represents the Intersection of sets (only the common elements).
  • A': Represents the Complement of set A (elements in the universal set U but not in A).
  • n(A): Represents the cardinal number of set A (the number of elements in the set).

4. What is the difference between the union (∪) and intersection (∩) of sets?

The main difference lies in how elements are combined. The Union (A ∪ B) of two sets A and B is a new set that contains all the elements from both A and B. The Intersection (A ∩ B) is a new set that contains only the elements that are common to both A and B. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5} while A ∩ B = {3}.

5. What is a Venn diagram and why is it useful for understanding sets?

A Venn diagram is a visual representation of sets using overlapping circles. Each circle represents a set, and the entire diagram is enclosed in a rectangle representing the universal set. They are useful because they provide a clear and intuitive way to see the relationships between sets, such as their intersection (the overlapping area) and union (all areas of the circles combined). This makes complex operations like A ∩ B or A ∪ B much easier to visualise.

6. How does the concept of a set apply to real-world situations?

Sets are a fundamental way of organising information and are used everywhere. For example:

  • In a kitchen, the collection of all spices is a set. The collection of all vegetables is another set.
  • In a school, the group of students in Class 11 is a set. The group of students in the cricket team is another set. The intersection would be Class 11 students who are also on the cricket team.
  • In computer science, databases use set theory to perform queries, filter data, and avoid duplicate entries.

7. What is the important distinction between a subset (⊆) and a proper subset (⊂)?

The distinction is about equality. A set A is a subset (⊆) of set B if every element of A is also in B. In this case, A can be equal to B. However, A is a proper subset (⊂) of B only if every element of A is in B, and there is at least one element in B that is not in A. In other words, for a proper subset, the two sets cannot be equal. For example, if A = {1, 2}, then {1, 2} is a subset of A, but it is not a proper subset of A.

8. Why is the empty set (∅) considered a subset of every set?

This is based on the formal definition of a subset. A set 'A' is a subset of set 'B' if there is no element in A that is not in B. Since the empty set has no elements, it's impossible to find an element in it that is not in set 'B'. Because this condition can never be proven false, the statement is considered universally true. Therefore, the empty set is a subset of any set, including itself.

9. How does the complement of a set depend on the universal set?

The complement of a set (A') is entirely dependent on the universal set (U). The complement A' is defined as the set of all elements that are in U but are not in A. If you change the universal set, the complement of A will also change. For example, if U = {1, 2, 3, 4, 5} and A = {1, 2}, then A' = {3, 4, 5}. But if U = {1, 2, 3}, then A' = {3}.

10. What is the key difference between 'equal sets' and 'equivalent sets'?

The difference is between the elements themselves versus the number of elements. Equal sets must have the exact same elements, although their order doesn't matter. For instance, {a, b, c} and {c, a, b} are equal sets. In contrast, equivalent sets simply need to have the same number of elements (the same cardinality), even if the elements are completely different. For example, {a, b, c} and {1, 2, 3} are equivalent sets because both have three elements, but they are not equal.