Set Theory is a branch of logical mathematics that studies the collection of objects and operations based on it.
The words collection, aggregate, and class are synonymous with set. On the other hand elements, members, and objects are synonymous and stand for the members of the set of which the set is comprised.
In this article, we will learn about the set theory and cover sets in detail. Look at the content guide that shows all the topics, we will be covering in this article.
Set Theory Definition
Sets are defined as ”a well-defined collection of objects”. Let’s say we have a set of Natural Numbers then it will have all the natural numbers as its members and the collection of the numbers is well defined that they are natural numbers.
Note: A set is always denoted by a capital letter.
A set of Natural Numbers is given by:
N = {1, 2, 3, 4…..}
The above example is a collection of natural numbers and is also well-defined. Well-defined means, that anyone should be able to tell whether the object belongs to the set or not.
Note:The term ‘well defined’ should be taken care of, as if we try to make a set of best players, then the term ‘best’ is not well defined. The concept of best, worst, beautiful, powerful, etc. varies according to the notions, assumptions, likes, dislikes, and biases of a person.
This explains what is a set, now let’s look at some set terms:
History of Set Theory
The concept of Set Theory was propounded in the year 1874 by Georg Cantor in his paper name ‘On a Property of Collection of All Real Algebraic Numbers‘.
His concept of Set Theory was later used by other mathematicians in giving various other theories such as Klein’s Encyclopedia and Russell Paradox. Sets Theory is a foundation for a better understanding of topology, abstract algebra, and discrete mathematics.
Understanding set theory will also help in understanding other mathematical concepts like relations, functions, probability, etc.
Examples of Sets
Some common examples of sets are mentioned below:
- Set of Natural Numbers: N = {1, 2, 3, 4….}
- Set of Even Numbers: E = {2, 4, 6, 8…}
- Set of Prime Numbers: P = {2, 3, 5, 7,….}
- Set of Integers: Z = {…, -4, -3, -2, -1, 0, 1, 2,….}
Some Common Sets
Below are the symbols used to represent common sets in set theory.
| Symbol | Set Name | Description | Example |
|---|---|---|---|
| N | Set of Natural Numbers | The set of all positive integers. Starting from 1. | {1, 2, 3, 4, …} |
| Z | Set of Integers | The set of all positive, negative, and zero integers. | {…, -3, -2, -1, 0, 1, 2, 3, …} |
| R | Set of Real Numbers | The set of all real numbers, including positive and negative rational and irrational numbers. | {…, -3.14, -2, -1, 0, 1, 2, 3.14, …} |
| C | Set of Complex Numbers | The set of all numbers that can be expressed as a + bi, where a and b are real numbers and i is the imaginary unit (√(-1)). | {1 + 2i, 3 – 5i, √2, -πi, …} |
| Q | Set of Rational Numbers | The set of all numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. | {1/2, -3/4, 5, 0.75, …} |
Important Terms Related to Set Theory
Some of the important terms related to sets are mentioned below. These terms will be used several times in this article, and knowing these terms will help you learn set theory.
Elements of a Set
The objects contained by a set are called the elements of the set.
They are represented using the ∈ symbol which means “belongs to”.
For Example:
In the set of Natural Numbers, 1, 2, 3, etc. are the objects, hence they are the elements of the set of Natural Numbers.
We can also say that 1 belongs to set N and it is represented as 1 ∈ N.
Cardinal Number of a Set
The number of elements present in a set is called the Cardinal Number of a Set.
For Example:
Suppose P is a set of the first five prime numbers given by P = {2, 3, 5, 7, 11}, then the Cardinal Number of set P is 5.
The Cardinal Number of Set P is represented by n(P) or |P| = 5.
Representation of Sets
Sets are primarily represented in two forms:
- Roster Form
- Set Builder Form
Roster Form
In the Roster Form of the set, the elements are placed inside braces {} and are separated by commas.
Let’s say we have a set of the first five prime numbers then it will be represented by P = {2, 3, 5, 7, 11}. Here the set P is an example of a finite set as the number of elements is finite, however, we can come across a set that has infinite elements then in that case the roster form is represented in the manner that some elements are placed followed by dots to represent infinity inside the braces.
Let’s say we have to represent a set of Natural Numbers in Roster Form then its Roster Form is given as N = {1, 2, 3, 4…..}.
In roster form representation, the set does not contain duplicate elements. For Example, If A represents a set that contains all letters of the word TREE, then the correct roster form representation will be:
A = {T,R,E} = {E,R,T}
A={T,R,E,E} is the wrong representation, therefore A≠ {T,R,E,E}
Set Builder Form
In Set Builder Form, a rule or a statement describing the common characteristics of all the elements is written instead of writing the elements directly inside the braces.
For Example, A set of all the prime numbers less than or equal to 10 is given as P = {p : p is a prime number ≤ 10}. In another example, the set of Natural Numbers in set builder form is given as N = {n : n is a natural number}.
Properties of Set Operations
The various properties followed by sets are tabulated below:
| Property | Expression |
|---|---|
| Commutative Property |
A ∪ B = B ∪ A A ∩ B = B ∩ A |
| Associative Property |
(A ∩ B) ∩ C = A ∩ (B ∩ C) (A ∪ B) ∪ C = A ∪ (B ∪ C) |
| Distributive Property |
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
| Identity Property |
A ∪ Φ = A A ∩ U = A |
| Complement Property | A ∪ A’ = U |
| Idempotent Property | A ∪ A = A ∩ A = A |
Set Theory Formulas
The set theory formulas are given for two sets – overlapping and disjoint sets. Both These formulas are given in detail in the article below:
De Morgan’s Laws
De Morgan’s Law is applicable in relating the union and intersection of two sets via their complements. There are two laws under De Morgan’s Law. Let’s learn them briefly
De Morgan’s Law of Union
De Morgan’s Law of Union states that the complement of the union of two sets is equal to the intersection of the complement of individual sets. Mathematically it can be expressed as
(A ∪ B)’ = A’ ∩ B’
De Morgan’s Law of Intersection
De Morgan’s Law of Intersection states that the complement of the intersection of two sets is equal to the union of the complement of individual sets. Mathematically it can be expressed as
(A ∩ B)’ = A’ ∪ B’
Visual Representation of Sets Using Venn Diagram
Venn Diagram is a technique for representing the relation between two sets with the help of circles, generally intersecting.
For Example:
Two circles intersecting with each other with the common area merged into them represent the union of sets, and two intersecting circles with a common area highlighted represent the intersection of sets while two circles separated from each other represent the two disjoint sets.
A rectangular box surrounding the circle represents the universal set. The Venn diagrams for various operations of sets are listed below:
Solved Examples on Set Theory
Example 1: If A and B are two sets such that n(A) = 17, n(B) = 23 and n(A ∪ B) = 38 then find n(A ∩ B).
Solution:
We know that n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
⇒ 38 = 17 + 23 – n(A ∩ B)
⇒ n(A ∩ B) = 40 – 38 = 2
Example 2: If X = {1, 2, 3, 4, 5}, Y = {4, 5, 6, 7, 8}, and Z = {7, 8, 9, 10, 11}, find (X ∪ Y), (X ∪ Z), (Y ∪ Z), (X ∪ Y ∪ Z), and X ∩ (Y ∪ Z)
Solution:
(X ∪ Y) = {1, 2, 3, 4, 5} ∪ {4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
(X ∪ Z) = {1, 2, 3, 4, 5} ∪ {7, 8, 9, 10, 11} = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}
(Y ∪ Z) = {4, 5, 6, 7, 8} ∪ {7, 8, 9, 10, 11} = {4, 5, 6, 7, 8, 9, 10, 11}
(X ∪ Y ∪ Z) = {1, 2, 3, 4, 5} ∪ {4, 5, 6, 7, 8} ∪ {7, 8, 9, 10, 11} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
X ∩ (Y ∪ Z) = {1, 2, 3, 4, 5} ∩ {4, 5, 6, 7, 8, 9, 10, 11} = {4, 5}
