Sets

Introduction

A set is a collection of well-defined objects that share common characteristics. In other words, a set is a group of objects with the same properties that helps to organise, interpret, and compare objects easily (such as a set of numbers, letters, or even real-life objects). The concept of sets is used everywhere, from elementary school math to computer science and real life. Therefore, it’s crucial to learn about types of sets, their representation, and formulas. In this article, we cover sets in detail, along with examples, types, and properties.

 

Table of Contents

• Sets Definition
• Representation of Sets
• Venn Diagrams and Sets
• Types of Sets
• Set Operations
• Sample Problems Based on Sets
• Practice Questions on Sets
• Conclusion
• FAQs on Sets

 

Sets Definition

A set is a well-defined group or collection of objects. These objects are called the elements or members of the set. The elements can be anything: numbers, letters, people, or even shapes.

What is Set?
A set is a collection of distinct elements, written within curly braces {} and separated by commas.

Example:
Set A = {2, 4, 6, 8}

This means set A contains 4 elements: 2, 4, 6, and 8.

What is Sets in Maths?

In mathematics, a set is defined as a collection of different objects grouped and represented using curly braces {}. The members of a set are called its elements. For example, the set of natural numbers is represented as:

N = {1, 2, 3, 4,...}, where 1, 2, 3 are the elements of the set.

Z = {...-3, -2, -1, 0, 1, 2, 3,...}, where Z represents the set of integers.

 

Representation of Sets

The sets are represented in curly braces. The elements of a set are usually represented in either statement form, roster form or set-builder form. Here are major ways and representation of sets in maths:

Standard Form: 

The standard or descriptive form is the way of defining a set without notations or listing elements. The set is defined with a description enclosed in curly braces. 

Example:

O = {odd numbers less than 10} is a set of odd numbers less than 10 defined in standard form.

Roster or Tabular Form:

Roster form or roster notation is a way of listing all elements of a set inside curly braces, separated by a comma. 

For Example: 

A = {1, 3, 5}, Listing all elements in curly brackets.

C = {c, o, f, f, e, e}, C is the set of all letters in the word coffee.

 

Set-Builder Form:

In set-builder form, a notation or a rule, or a condition is given to define the elements. We use a variable for instance x to write the condition after a vertical bar ( | ), which is read as “such that.”

Example: 

A = {x | x is an odd number less than 6}, describing elements using a rule.

B = {x | x is a natural number less than 5}

This means: B = {1, 2, 3, 4}

 

Venn Diagrams and Sets

Venn diagrams help visualize sets and their relationships. It is a graphical method used to show relationships between sets.

• A single circle shows one set.

• Overlapping circles show common elements.

• Non-overlapping circles show disjoint sets.

Example:
Let A = {1, 2, 3}, B = {3, 4, 5}

The common element is 3; it appears in the overlapping region.

 

Symbols Used in Sets

There are different symbols used in sets to represent union (∪), intersection (∩), subset (⊆), superset (⊇), and more. Here are some common symbols used while working with sets:

 

 

Types of Sets

To truly define sets, we must understand their types:

Finite & Infinite Sets

Finite Set: A finite set is a set that has limited number of elements.

For Example: A= {a, e, i, o, u}

Infinite Set: Infinite set is a set that goes on forever, i.e., there are infinite number of elements in an infinite set.

For Example: N = {1, 2, 3, 4, ...}

Equal and Equivalent Sets

Two sets are called equal sets if there are same elements in each of them.
Example: A = {1, 2, 4} and B = {2, 4, 1} → A = B

Equivalent sets have the same number of elements but may differ in content.  

Example: A = {apple, banana}, B = {red, blue}  

Subset and Power Sets  

A set formed from elements of another set is known as a Subset.
If B = {2, 4} and A = {2, 4, 6}, then B ⊂ A

A power set is the set of all possible subsets of a given set.
If A = {1, 2} then power set of A denoted by P(A) = {{}, {1}, {2}, {1, 2}}

Empty or Null Set

A set with no elements or zero elements is called an empty or null set.
Example: A = {} or A = ∅

Singleton Set

A set with only one element is known as a singleton set.
Example: A = {0}

Universal Set

A set that contains all elements under consideration is called a universal set.
Example: If discussing numbers 1 to 10, then U = {1, 2, ..., 10}

 

Set Operations

Here are the main operations you can perform with sets:

Union (∪) of Sets 

Union of two sets means putting together all elements from both sets, without repeating any element. Think of it as “everything in A, everything in B, and anything they both have.”

Union of set A and B, is denoted byA∪B.

Example:
A = {1, 2}, B = {2, 3}
A ∪ B = {1, 2, 3}

Intersection (∩) of Sets

Intersection of sets is a set that contains only the common elements present in both sets. 

Union of set A and B, is denoted by (A ∩ B).

Example:
A = {1, 2, 3}, B = {2, 3, 4}
A ∩ B = {2, 3}

Difference of Sets (A - B)

Difference of set A and B is a set with elements that are present in A but not B.

Example:
A = {1, 2, 3}, B = {2, 3, 4}
A - B = {1}

Complement of a Set (A’)

Complement of a set is set that contains the elements that are present not in the set but in the universal set.

If U = {1 to 5}, A = {2, 3},
Then A’ = {1, 4, 5}

 

Sample Problems Based on Sets

Let’s go through some practice problems to build understanding of sets and their operations:

Problem 1: Write the set A = {2, 4, 6, 8, 10} in set-builder form.

Solution: To write set A = {2, 4, 6, 8, 10} in set-builder form follow the steps below:

Step 1: Observe the pattern of elements. They are even natural numbers up to 10.

Step 2: Write the condition in set-builder form using variable x:

 A = { x | x is an even natural number, 2 ≤ x ≤ 10 } 

Problem 2: Write A = {a, e, i, o, u} in set-builder form.

Solution: A = {a, e, i, o, u}

Set-builder form:
𝐴={ 𝑥 ∣ 𝑥 is a vowel in the English alphabet}

Problem 3: Write the subsets of A = {2,4,6}.

Solution: 

The subsets of A are:

φ, {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {2, 4, 6}

Problem 4: Write the set of letters in the word “COFFEE”.

Solution: A = {C, O, F, F, E, E}
Each element appears only once because sets do not allow repetition.

Problem 5: Find the union A∪B where A = {12, 22}, B = {20, 30}?

Solution: Set A = {12, 22} and set B = {20, 30}

To find the union of sets A and B, we have to write all elements A and B together as:

A∪B = {12, 22, 20, 30}

Problem 6: Find the intersection of A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8, 10}?

Solution: Set A = {1, 2, 3, 4, 5} and set B = {2, 4, 6, 8, 10}

To find the intersection of sets A and B, we have to write all common elements from set A and B as:

A ∩ B = {2, 4}

 

Practice Questions on Sets

You can practice the following set questions on your own:  

1. Write the set of vowels in English.

2. List even numbers from 1 to 10 in set-builder form.

3. Find A ∩ B for A = {1, 3, 5}, B = {3, 4, 5}

4. If U = {1 to 10}, A = {2, 4, 6}, find A’

5. Create a Venn diagram showing A = {1, 2}, B = {2, 3}

 

Conclusion

While sets are just one concept in mathematics to learn, it is impossible to succeed in the subject without understanding them. Learning how to define sets, their types, union, intersection and formulas is essential not only for math but other subjects too. Venn diagrams used to represent the union or intersection of sets, prepares us in the skill to learn higher mathematics and work on different tasks independently.

 

Frequently Asked Questions On Sets

1. What do you mean by sets?

Ans: In mathematics, a set is a collection of distinct and well-defined objects or elements. These objects can be numbers, letters, or even items, and they are usually written inside curly brackets like {1, 2, 3}.

 

2. What is the universal set 1 2 3 4 5 6 7 8 9 10?

Ans: The universal set includes all elements under consideration for a particular discussion. If we are talking about numbers from 1 to 10, then the universal set is written as:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

 

3. What are the 12 types of sets?

Ans: Here are 12 types of sets in mathematics:

1. Empty Set

2. Finite Set

3. Infinite Set

4. Equal Set

5. Unequal Set

6. Subset

7. Proper Subset

8. Improper Subset

9. Universal Set

10. Power Set

11. Singleton Set

12. Disjoint Set

 

4. What is the Principle of inclusion-exclusion for two sets?

Ans: The Principle of Inclusion-Exclusion for two sets or Union-Intersection formula is used to find the total number of elements in two overlapping sets A and B.
n(A∪B) = n(A) + n(B) - n(A ∩ B)

5. What do you mean by 5 sets?

Ans: "5 sets" usually means a group of 5 different sets. For example:
Set A = {1, 2}
Set B = {3, 4}
Set C = {5}
Set D = {6, 7, 8}
Set E = {9, 10}
These are five different sets, each with its own elements

 

Want to learn more about how sets connect with geometry, algebra, and real-world data?
Explore more math concepts at Orchids The International School.