Sets

Sets in mathematics, are simply a collection of distinct objects forming a group. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. Every item in the set is called an element of the set. Curly brackets are used while writing a set. A very simple example of a set would be like this. Set A = {1, 2, 3, 4, 5}. In set theory, there are various notations to represent elements of a set. Sets are usually represented using a roster form or a set builder form. Let us discuss each of these terms in detail.

In mathematics, a set is defined as a well-defined collection of objects. Sets are named and represented using capital letters. In the set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.

Sets in Maths Examples

Some standard sets in maths are:

• Set of natural numbers, ℕ = {1, 2, 3, ...}
• Set of whole numbers, W = {0, 1, 2, 3, ...}
• Set of integers, ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Set of rational numbers, ℚ = {p/q | q is an integer and q ≠ 0}
• Set of irrational numbers, ℚ' = {x | x is not rational}
• Set of real numbers, ℝ = ℚ ∪ ℚ'

All these are infinite sets. But there can be finite sets as well. For example, the collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}, which is a finite set.

Let us use this example to understand the basic terminology associated with sets in math.

Elements of a Set

The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas. To denote that an element is contained in a set, the symbol '∈' is used. In the above example, 2 ∈ A. If an element is not a member of a set, then it is denoted using the symbol '∉'. For example, 3 ∉ A.

Cardinal Number of a Set

The cardinal number, cardinality, or order of a set denotes the total number of elements in the set. For natural even numbers less than 10, n(A) = 4. Sets are defined as a collection of unique elements. One important condition to define a set is that all the elements of a set should be related to each other and share a common property. For example, if we define a set with the elements as the names of months in a year, then we can say that all the elements of the set are the months of the year.

There are different set notations used for the representation of sets in set theory. They differ in the way in which the elements are listed. The three set notations used for representing sets are:

• Semantic form
• Roster form
• Set builder form

Let us understand each of these forms with an example.

Semantic Form

Semantic notation describes a statement to show what are the elements of a set. For example, a set of the first five odd numbers.

Roster Form

The most common form used to represent sets is the roster notation in which the elements of the sets are enclosed in curly brackets separated by commas. For example, Set B = {2,4,6,8,10}, which is the collection of the first five even numbers. In a roster form, the order of the elements of the set does not matter, for example, the set of the first five even numbers can also be defined as {2,6,8,10,4}. Also, if there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element. For example, infinite sets are represented as, X = {1, 2, 3, 4, 5 ...}, where X is the set of natural numbers. To sum up the notation of the roster form, please take a look at the examples below.

Finite Roster Notation of Sets : Set A = {1, 2, 3, 4, 5} (The first five natural numbers)

Infinite Roster Notation of Sets : Set B = {5, 10, 15, 20 ....} (The multiples of 5)

Set Builder Form

The set builder notation has a certain rule or a statement that specifically describes the common feature of all the elements of a set. The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set. For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20. Sometimes a ":" is used in the place of the "|".

Visual Representation of Sets Using Venn Diagram

Venn Diagram is a pictorial representation of sets, with each set represented as a circle. The elements of a set are present inside the circles. Sometimes a rectangle encloses the circles, which represents the universal set. The Venn diagram represents how the given sets are related to each other.

Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning.

There are different types of sets in set theory. Some of these are singleton, finite, infinite, empty, etc.

Singleton Sets

A set that has only one element is called a singleton set or also called a unit set. Example, Set A = { k | k is an integer between 3 and 5} which is A = {4}.

Finite Sets

As the name implies, a set with a finite or countable number of elements is called a finite set. Example, Set B = {k | k is a prime number less than 20}, which is B = {2,3,5,7,11,13,17,19}

Infinite Sets

A set with an infinite number of elements is called an infinite set. Example: Set C = {Multiples of 3}.

Empty or Null Sets

A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as 'phi'. Example: Set X = { }.

Equal Sets

If two sets have the same elements in them, then they are called equal sets. Example: A = {1,2,3} and B = {1,2,3}. Here, set A and set B are equal sets. This can be represented as A = B.

Unequal Sets

If two sets have at least one different element, then they are unequal sets. Example: A = {1,2,3} and B = {2,3,4}. Here, set A and set B are unequal sets. This can be represented as A ≠ B.

Equivalent Sets

Two sets are said to be equivalent sets when they have the same number of elements, though the elements are different. Example: A = {1,2,3,4} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)

Overlapping Sets

Two sets are said to be overlapping if at least one element from set A is present in set B. Example: A = {2,4,6} B = {4,8,10}. Here, element 4 is present in set A as well as in set B. Therefore, A and B are overlapping sets.

Disjoint Sets

Two sets are disjoint if there are no common elements in both sets. Example: A = {1,2,3,4} B = {5,6,7,8}. Here, set A and set B are disjoint sets.

Subset and Superset

For two sets A and B, if every element in set A is present in set B, then set A is a subset of set B(A ⊆ B) and in this case, B is the superset of set A(B ⊇ A).
Example: Consider the sets A = {1,2,3} and B = {1,2,3,4,5,6}. Here:

• A ⊆ B, since all the elements in set A are present in set B.
• B ⊇ A denotes that set B is the superset of set A.

Universal Set

A universal set is the collection of all the elements regarding a particular subject. The universal set is denoted by the letter 'U'. Example: Let U = {The list of all road transport vehicles}. Here, a set of cars is a subset for this universal set, the set of cycles, trains are all subsets of this universal set.

Power Sets

Power set is the set of all subsets that a set could contain. Example: Set A = {1,2,3}. Power set of A is = {∅, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

Some important operations on sets in set theory include union, intersection, difference, the complement of a set, and the cartesian product of a set. A brief explanation of set operations is as follows.

Union of Sets

Union of sets, which is denoted as A U B, lists the elements in set A and set B or the elements in both set A and set B. For example, {1, 3} ∪ {1, 4} = {1, 3, 4}

Intersection of Sets

The intersection of sets which is denoted by A ∩ B lists the elements that are common to both set A and set B. For example, {1, 2} ∩ {2, 4} = {2}

Set Difference

Set difference which is denoted by A - B, lists the elements in set A that are not present in set B. For example, A = {2, 3, 4} and B = {4, 5, 6}. A - B = {2, 3}.

Set Complement

Set complement which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.

Cartesian Product of Sets

The cartesian product of two sets which is denoted by A × B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained. For example, {1, 3} × {1, 3} = {(1, 1), (1, 3), (3, 1), (3, 3)}.

In the above figure, the shaded portions in "blue" show the set that they are labelled with.

Sets find their application in the field of algebra, statistics, and probability. There are some important set theory formulas in set theory as listed below.

For any two overlapping sets A and B,

• n(A U B) = n(A) + n(B) - n(A ∩ B)
• n (A ∩ B) = n(A) + n(B) - n(A U B)
• n(A) = n(A U B) + n(A ∩ B) - n(B)
• n(B) = n(A U B) + n(A ∩ B) - n(A)
• n(A - B) = n(A U B) - n(B)
• n(A - B) = n(A) - n(A ∩ B)

For any two sets A and B that are disjoint,

• n(A U B) = n(A) + n(B)
• A ∩ B = ∅
• n(A - B) = n(A)

Similar to numbers, sets also have properties like associative property, commutative property, and so on. There are six important properties of sets. Given, three sets A, B, and C, the properties for these sets are as follows.

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What is Set in Math?

Sets are a collection of distinct elements, which are enclosed in curly brackets, separated by commas. The list of items in a set is called the elements of a set. Examples are a collection of fruits, a collection of pictures. Sets are represented by the symbol { }. i.e., the elements of the set are written inside these brackets. Example: Set A = {a,b,c,d}. Here, a,b,c, and d are the elements of set A.

What are Different Sets Notations to Represent Sets?

Sets can be represented in three ways. Representing sets means a way of listing the elements of the set. They are as follows.

• Semantic Notation: The elements of a set are represented by a single statement. For example, Set A is the number of days in a week.
• Roster Notation: This form of representation of sets uses curly brackets to list the elements of the set. For example, Set A = {10,12,14,16,18}}
• Set Builder Notation: A set builder form represents the elements of a set by a common rule or a property. For example, {x | x is a prime number less than 20}

What are the Types of Sets?

Sets differ from each depending upon elements present in them. Based on this, we have the following types of sets. They are singleton sets, finite and infinite sets, empty or null sets, equal sets, unequal sets, equivalent sets, overlapping sets, disjoint sets, subsets, supersets, power sets, and universal sets.

What are the Properties of Sets in Set-Theory?

Different properties associated with sets in math are,

• Commutative Property: A U B = B U A and A ∩ B = B ∩ A
• Associative Property: (A ∩ B) ∩ C = A ∩ (B ∩ C) and (A U B) U C = A U (B U C)
• Distributive Property: A U (B ∩ C) = (A U B) ∩ (A U C) and A ∩ (B U C) = (A ∩ B) U (A ∩ C)
• Identity Property: A U ∅ = A and A ∩ U = A
• Complement Property: A U A' = U
• Idempotent Property: A ∩ A = A and A U A = A

What is the Union of Sets?

The union of two sets A and B are the elements from both set A and B, or both combined together. It is denoted using the symbol 'U'. For example, if set A = {1,2,3} and set B = {4,5,6}, then A U B = {1,2,3,4,5,6}. A U B is read as 'A union B'.

What is the Intersection of Sets?

The intersection of two sets A and B are the elements that are common to both set A and B. It is denoted using the symbol '∩'. For example, if set A = {1,2,3} and set B = {3,4,5}, then A ∩ B = {3}. A ∩ B is read as 'A intersection B'.

What are Subsets and Supersets?

If every element in a set A is present in set B, then set B is the superset of set A and set A is a subset of set B. Example: A = {1,4,5} B = {1,2,3,4,5,6}, here since all elements of set A are present in set B ⇒ A ⊆ B and B ⊇ A.

What are Universal Sets?

A universal set, denoted by the letter 'U', is the collection of all the elements in regard to a particular subject. Example: Let U = {All types of cycles}. Here, a set of cycles of a specific company is a subset of this universal set.

What Does Sets Class 11 Contain?

The sets in class 11 is an important chapter that deals with various components of set theory. It starts with definition of sets, and extends to types of sets, properties of sets, set operations, etc. It also has some real-life applications related to sets. To solve more applications related to sets class 11, click here.
☛Also Check:

• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.1
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.2
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.3
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.4
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.5
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.6
• NCERT Solutions Class 11 Maths Chapter 1 Miscellaneous Exercise

What is Complement in Sets?

The complement of a set which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.

What is Cartesian Product in Sets?

Cartesian product of two sets, denoted by A×B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained. For example, if A = {1,2} and B = {3,4}, then A×B = {(1,3), (1,4), (2,3), (2,4)}.

What is the Use of Venn Diagram in Set Theory?

Venn Diagram is a pictorial representation of the relationship between two or more sets. Circles are used to represent sets. Each circle represents a set. A rectangle that encloses the circles represents the universal set.