Sets in Mathematics: Definition, Equation, Formula, Examples

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. Roster and Set Builder sets are the basic elemental mathematical concepts that one could use to explain real-life examples of the said topic. Consider a typical classroom scenario: suppose only a teacher who intends to classify learners based on their preferences in class contents would need such a relationship. Through this, she names a set of all the students who have an interest in mathematics to ensure she only tests the skills and knowledge of those who have an interest in the subject.

This article will cover the concept of the sets and their different forms. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

Sets

Definition of Set: A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V. $\qquad$
(I) The set includes objects, members, and elements and all are equal terms
(ii) The sets are denoted by $\mathrm{A}, \mathrm{B}, \mathrm{C}$, etc.
(iii) The elements of a set are represented by small letters a, b, c, $x, y, z$, etc.

If $a$ is an element of a set $A$, we say that " a belongs to $A$ " The Greek symbol $E$ (epsilon) is used to denote the phrase 'belongs to'.

Example: $A=\{1,2,3\}$
All the objects that form a set are called its elements or members.
Example: $A=\{1,2,3\}$, then $2 \in A$ ( 2 belongs to set $A)$ and $4 \notin A$ ( 4 does not belong to set A)

Therefore, We shall say that a set is a well-defined collection of objects.
There are two methods of representing a set :
(i) Roster or tabular form
(ii) Set-builder form.

Properties of Sets:

1. A set is well-defined if it is possible to determine if the object belongs to the set or not.

2. It is unordered If it's not in the order. For example, the set $\{1,2,3\}$ is the same as $\{3,2,1\}$.
3. A set cannot have duplicate elements.

Types of Sets

1. Empty Set: A set that does not contain any element is called an empty set void set or null set. It is denoted by $\varnothing$.

Example: A set of mangoes in the basket of guavas is an example of an empty set because, in a guava basket, there is no mango present.

2. Singleton Set: A set which is having only one element is called a singleton set.

Example: There is only one apple in a basket of grapes.

3. Finite set: A set which is empty or consists of a finite number of elements is called a finite set.

Example: A set of natural numbers up to 10.

$
A=\{1,2,3,4,5,6,7,8,9,10\}
$

4. Infinite set: A set which has infinite elements is called an infinite set.

Example: A set of all-natural numbers.

$
A=\{1,2,3,4,5,6,7,8,9 \ldots \ldots\}
$

5. Equivalent set: Two sets having the same number of elements are called equivalent sets. It is represented as:

$
n(A)=n(B)
$

where $A$ and $B$ are two different sets with the same number of elements.
Example: If $A=\{1,2,3,4\}$ and $B=\{$ Red, Blue, Green, Black $\}$
In set $A$, there are four elements and in set $B$ also there are four elements. Therefore, set $A$ and set $B$ are equivalent.

6. Equal sets: Equal sets contain the same or equal elements as the other and they suggest that there is a one-to-one relationship in which each member of one of the sets is also a member of the other without duplication or omission of any element.

Example: $A=\{1,2,3,4\}$ and $B=\{4,3,2,1\}$

$
A=B
$

7. Disjoint Sets: The two sets A and B are said to be disjoint if the set does not contain any common element.

Example: Set $A=\{1,2,3,4\}$ and set $B=\{5,6,7,8\}$ are disjoint sets because there is no common element between them.

Sets Operations

1. Union: The union of $A$ and $B$ is the set that consists of all the elements of $A$ and all the elements of $B$, the common elements being taken only once. The symbol ' $u$ ' is used to denote the union. Symbolically, we write $A$ $U B=\{x: x \in A$ or $x \in B\}$

2. Intersection: The intersection of sets $A$ and $B$ is the set of all elements that are common to both $A$ and $B$. The symbol ' $\cap$ 'is used to denote the intersection. Symbolically, we write $A \cap B=\{x: x \in A$ and $x \in B\}$

3. Difference: The difference of the sets $A$ and $B$ in this order is the set of elements that belong to A but not to B. Symbolically, we write A - B and read as "A minus B".

4. Complement: Let $U$ be the universal set and $A$ is a subset of $U$. Then the complement of $A$ is the set of all elements of $U$ which are not the elements of A. Symbolically, we use $A^{\prime}$ or $A^C$ to denote the complement of $A$ with respect to $U . A^{\prime}=\{x: x \in U$ and $x \notin A\}$. Obviously, $A^{\prime}=U-A$

5. Cartesian Product: The Cartesian product of two sets $A$ and $B$, denoted by $A \times B$ is the set of all ordered pairs ( $a, b$ ) where $a$ is an element of $A$ and $b$ is an element of $B$.

Solved Examples Based on the Sets

1. Which of the following is a set?

1) The list of all the bright colours.

2) The list of all the dull colours.

3) The list of all colours in the Rainbow.

4) The list of all the good colours.

Solution:

As we learned

A set is a well-defined collection of objects. eg. $A=\{1,2,3\}$.
In this Question,
Bright, dull, and good colours are not well-defined as it is different for different people. But, the list of all colours in the rainbow is definite and well-defined. So, it is a set.

We can decide with respect to any color, say green, whether it will lie in the set or not. So, it is a well-defined collection. We cannot do this in case of bright, dull or good colours.

Hence, the answer is the option 3.
Example 2: Which of the following sets has an infinite number of elements?
1) $A=\{x: x$ is an odd number on dice $\}$
2) $B=\{x: x$ is a prime number $\}$
3) $C=\{x: x$ is a factor of 24$\}$
4) $D=\{x: x$ is an even prime number $\}$

Solution

Option $1=\{1,3,5\}$ : so finite number of elements.
Option $2=\{2,3,5,7, \ldots .$.$\} : so infinite number of elements.$
Option $3=\{1,2,3,4,6,8,12,24\}$ : so finite number of elements.
Option $4=\{2\}$ : so finite number of elements.
Hence, the answer is the option 2.
Example 3 : Which of the following sets is different from the other three?
1) $A=\{x: x$ is odd, $x \in Z\}$
2) $B=\{x: x$ is not divisible by $2, x \in Z\}$
3) $C=\{x: x$ is the half of an even integer $\}$
4) $D=\{x: x$ is not even, $x \in Z\}$

Solution

Half of an even integer can be even as well as an odd integer. Eg: half of 4 is 2 .

All other options denote odd integers.
So, 3 is different from others.
Hence, the answer is the option 3.
Example 4: Which of the following is not a set?
1) The collection of all licensed drivers in the class.
2) The collection of students in a class above the age of 15.
3) The collection of all the young students in the class.
4) The collection of all students with names starting from ' $A$ '.

Solution

As we learned

A set is a well-defined collection of objects.

In this question,

"The collection of young students" is not a set because the term young is not well defined.

In all other options, we can identify the elements present in those collections, so they are sets.

Hence, the answer is the option 3.

Example 5: If $\operatorname{set} A=\{x \mid x$ is the square of a prime number $\}$, then which of the following is an element of set $A$?
1) 64
2) 961
3) 729
4) 625

Solution

Since 961 is the square of 31, which is prime, so 961 is an element of $\operatorname{set} A$.

Hence, the answer is the option 2.

Summary

Sets play a crucial role in both computational and theoretical contexts due to their fundamental nature in mathematics, which makes defining collections of objects, performing operations like unions and intersections, and establishing relationships more efficient. For example, concepts such as functions and probability frequently rely on sets to simplify and structure calculations. The unique structure and properties of sets make them suitable for use in various fields like computer science, statistics, and logic.