## Chapter 1 Sets

- In these days of conflict between ancient and modern studies; there must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but is the oldest and the youngest. - G.H. HARDY

### 1.1 Introduction

The concept of set serves as a fundamental part of the present day mathematics. Today this concept is being used in almost every branch of mathematics. Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets.

The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”. In this Chapter, we discuss some basic definitions and operations involving sets.

### 1.2 Sets and their Representations

In everyday life, we often speak of collections of objects of a particular kind, such as, a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come across collections, for example, of natural numbers, points, prime numbers, etc. More specially, we examine the following collections:

(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9

(ii) The rivers of India

(iii) The vowels in the English alphabet, namely, $a, e, i, o, u$

(iv) Various kinds of triangles

(v) Prime factors of 210, namely, 2,3,5 and 7

(vi) The solution of the equation: $x^{2}-5 x+6=0$, viz, 2 and 3 .

We note that each of the above example is a well-defined collection of objects in the sense that we can definitely decide whether a given particular object belongs to a given collection or not. For example, we can say that the river Nile does not belong to the collection of rivers of India. On the other hand, the river Ganga does belong to this colleciton.

We give below a few more examples of sets used particularly in mathematics, viz.

$\mathbf{N}$ : the set of all natural numbers

$\mathbf{Z}$ : the set of all integers

$\mathbf{Q}$ : the set of all rational numbers

$\mathbf{R}$ : the set of real numbers

$\mathbf{Z^{+}} $: the set of positive integers

$\mathbf{Q^{+}} $: the set of positive rational numbers, and

$\mathbf{R^{+}} $: the set of positive real numbers.

The symbols for the special sets given above will be referred to throughout this text.

Again the collection of five most renowned mathematicians of the world is not well-defined, because the criterion for determining a mathematician as most renowned may vary from person to person. Thus, it is not a well-defined collection.

We shall say that **a set is a well-defined collection of objects**.

The following points may be noted :

(i) Objects, elements and members of a set are synonymous terms.

(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, 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 $\in$ (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write $a \in A$. If ’ $b$ ’ is not an element of a set $A$, we write $b \notin A$ and read " $b$ does not belong to A".

Thus, in the set $V$ of vowels in the English alphabet, $a \in V$ but $b \notin V$. In the set $P$ of prime factors of $30,3 \in P$ but $15 \notin P$.

There are two methods of representing a set :

(i) Roster or tabular form

(ii) Set-builder form.

(i) In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { } . For example, the set of all even positive integers less than 7 is described in roster form as $\{2,4,6\}$. Some more examples of representing a set in roster form are given below :

(a) The set of all natural numbers which divide 42 is $\{1,2,3,6,7,14,21,42\}$.

Note - In roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as $\{1,3,7,21,2,6,14,42\}$.

(b) The set of all vowels in the English alphabet is $\{a, e, i, o, u\}$.

(c) The set of odd natural numbers is represented by $\{1,3,5, \ldots\}$. The dots tell us that the list of odd numbers continue indefinitely.

Note - It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct. For example, the set of letters forming the word ‘SCHOOL’ is $\{S, C, H, O, L\}$ or $\{H, O, L, C, S\}$. Here, the order of listing elements has no relevance.

(ii) In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set $\{a, e, i, o, u\}$, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by $V$, we write

$V=\{x: x$ is a vowel in English alphabet $\}$

It may be observed that we describe the element of the set by using a symbol $x$ (any other symbol like the letters $y, z$, etc. could be used) which is followed by a colon " : “. After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces. The above description of the set $V$ is read as “the set of all $x$ such that $x$ is a vowel of the English alphabet”. In this description the braces stand for “the set of all”, the colon stands for “such that”. For example, the set

$A=\{x: x$ is a natural number and $3<x<10\}$ is read as “the set of all $x$ such that $x$ is a natural number and $x$ lies between 3 and 10.” Hence, the numbers 4, 5, 6, 7,8 and 9 are the elements of the set $A$.

If we denote the sets described in $(a),(b)$ and $(c)$ above in roster form by $A, B$, $C$, respectively, then $A, B, C$ can also be represented in set-builder form as follows:

$A=\{x: x$ is a natural number which divides 42 $\}$

$B=\{y: y$ is a vowel in the English alphabet $\}$

$C=\{z: z$ is an odd natural number $\}$

### 1.3 The Empty Set

Consider the set

$A=\{x: x$ is a student of Class XI presently studying in a school $\}$

We can go to the school and count the number of students presently studying in Class XI in the school. Thus, the set A contains a finite number of elements.

We now write another set $B$ as follows:

$B = \{x: x$ is a student presently studying in both Classes X and XI $\}$

We observe that a student cannot study simultaneously in both Classes X and XI. Thus, the set B contains no element at all.

**Definition 1** A set which does not contain any element is called the empty set or the null set or the void set.

According to this definition, B is an empty set while A is not an empty set. The empty set is denoted by the symbol $\phi$ or { }.

We give below a few examples of empty sets.

(i) Let $A=\{x: 1<x<2, x$ is a natural number $\}$. Then A is the empty set, because there is no natural number between 1 and 2.

(ii) $B=\{x: x^{2}-2=0$ and $x$ is rational number $\}$. Then $B$ is the empty set because the equation $x^{2}-2=0$ is not satisfied by any rational value of $x$.

(iii) $C =$ $\{x: x$ is an even prime number greater than 2 $\}$. Then $C$ is the empty set, because 2 is the only even prime number.

(iv) $D=\{x: x^{2}=4, x.$ is odd $\}$. Then $D$ is the empty set, because the equation $x^{2}=4$ is not satisfied by any odd value of $x$.

### 1.4 Finite and Infinite Sets

Let $\quad A=\{1,2,3,4,5\}, \quad B=\{a, b, c, d, e, g\}$

and $\quad C=\{$ men living presently in different parts of the world $\}$

We observe that A contains 5 elements and B contains 6 elements. How many elements does $C$ contain? As it is, we do not know the number of elements in $C$, but it is some natural number which may be quite a big number. By number of elements of a set $S$, we mean the number of distinct elements of the set and we denote it by $n$ (S). If $n$ (S) is a natural number, then $S$ is non-empty finite set.

Consider the set of natural numbers. We see that the number of elements of this set is not finite since there are infinite number of natural numbers. We say that the set of natural numbers is an infinite set. The sets A, B and C given above are finite sets and $n(A)=5, n(B)=6$ and $n(C)=$ some finite number.

**Definition 2** A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.

Consider some examples :

(i) Let $W$ be the set of the days of the week. Then $W$ is finite.

(ii) Let $S$ be the set of solutions of the equation $x^{2}-16=0$. Then $S$ is finite.

(iii) Let $G$ be the set of points on a line. Then $G$ is infinite.

When we represent a set in the roster form, we write all the elements of the set within braces { }. It is not possible to write all the elements of an infinite set within braces { } because the numbers of elements of such a set is not finite. So, we represent some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed ( or preceded ) by three dots.

For example, $\{1,2,3 \ldots\}$ is the set of natural numbers, $\{1,3,5,7, \ldots\}$ is the set of odd natural numbers, $\{ \ldots,-3,-2,-1,0,1,2,3, \ldots\}$ is the set of integers. All these sets are infinite.

Note - All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

### 1.5 Equal Sets

Given two sets A and B, if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal. Clearly, the two sets have exactly the same elements.

**Definition 3** Two sets A and B are said to be equal if they have exactly the same elements and we write $A=B$. Otherwise, the sets are said to be unequal and we write $A \neq B$.

We consider the following examples :

(i) Let $A=\{1,2,3,4\}$ and $B=\{3,1,4,2\}$. Then $A=B$.

(ii) Let $A$ be the set of prime numbers less than 6 and $P$ the set of prime factors of 30. Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6 .

Note - A set does not change if one or more elements of the set are repeated. For example, the sets $A=\{1,2,3\}$ and $B=\{2,2,1,3,3\}$ are equal, since each element of A is in B and vice-versa. That is why we generally do not repeat any element in describing a set.

### 1.6 Subsets

Consider the sets : $X=$ set of all students in your school, $Y=$ set of all students in your class.

We note that every element of $Y$ is also an element of $X$; we say that $Y$ is a subset of $X$. The fact that $Y$ is subset of $X$ is expressed in symbols as $Y \subset X$. The symbol $\subset$ stands for ‘is a subset of’ or ‘is contained in’.

**Definition 4** $A$ set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of B.

In other words, $A \subset B$ if whenever $a \in A$, then $a \in B$. It is often convenient to use the symbol " $\Rightarrow$ " which means implies. Using this symbol, we can write the definiton of subset as follows:

$$ A \subset B \text { if } a \in A \Rightarrow a \in B $$

We read the above statement as " $A$ is a subset of $B$ if $a$ is an element of $A$ implies that $a$ is also an element of $B$ “. If $A$ is not a subset of $B$, we write $A \not \subset B$.

We may note that for $A$ to be a subset of $B$, all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A. If it so happens that every element of $B$ is also in $A$, then we shall also have $B \subset A$. In this case, $A$ and $B$ are the same sets so that we have $A \subset B$ and $B \subset A \Leftrightarrow A=B$, where " $\Leftrightarrow$ " is a symbol for two way implications, and is usually read as if and only if (briefly written as “iff”).

It follows from the above definition that every set $A$ is a subset of itself, i.e., $A \subset A$. Since the empty set $\phi$ has no elements, we agree to say that $\phi$ is a subset of every set. We now consider some examples :

(i) The set $\mathbf{Q}$ of rational numbers is a subset of the set $\mathbf{R}$ of real numbes, and we write $\mathbf{Q} \subset R$.

(ii) If $A$ is the set of all divisors of 56 and $B$ the set of all prime divisors of 56, then $B$ is a subset of $A$ and we write $B \subset A$.

(iii) Let $A=\{1,3,5\}$ and $B=\{x: x$ is an odd natural number less than 6 $\}$. Then $A \subset B$ and $B \subset A$ and hence $A=B$.

(iv) Let $A=\{a, e, i, o, u\}$ and $B=\{a, b, c, d\}$. Then $A$ is not a subset of $B$, also $B$ is not a subset of $A$.

Let $A$ and $B$ be two sets. If $A \subset B$ and $A \neq B$, then $A$ is called a proper subset of $B$ and $B$ is called superset of $A$. For example,

$A=\{1,2,3\}$ is a proper subset of $B=\{1,2,3,4\}$.

If a set $A$ has only one element, we call it a singleton set. Thus, $\{a\}$ is a singleton set.

#### 1.6.1 Subsets of set of real numbers

As noted in Section 1.6, there are many important subsets of $\mathbf{R}$. We give below the names of some of these subsets.

The set of natural numbers $\mathbf{N}=\{1,2,3,4,5, \ldots\}$

The set of integers $\quad \mathbf{Z}=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\}$

The set of rational numbers $\mathbf{Q}=\{x: x=\frac{p}{q}, p, q \in \mathbf{Z}$ and $q \neq 0\}$ which is read " $\mathbf{Q}$ is the set of all numbers $x$ such that $x$ equals the quotient $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero”. Members of $\mathbf{Q}$ include -5 (which can be expressed as $-\frac{5}{1}$ ), $\frac{5}{7}, 3 \frac{1}{2} \quad$ (which can be expressed as $\frac{7}{2}$ ) and $-\frac{11}{3}$.

The set of irrational numbers, denoted by $\mathbf{T}$, is composed of all other real numbers. Thus $\quad \mathbf{T}=\{x: x \in \mathbf{R}$ and $x \notin \mathbf{Q}\}$, i.e., all real numbers that are not rational. Members of $\mathbf{T}$ include $\sqrt{2}, \sqrt{5}$ and $\pi$.

Some of the obvious relations among these subsets are:

$$ \mathbf{N} \subset \mathbf{Z} \subset \mathbf{Q}, \mathbf{Q} \subset \mathbf{R}, \mathbf{T} \subset \mathbf{R}, \mathbf{N} \not \subset \mathbf{T} . $$

#### 1.6.2 Intervals as subsets of $\mathbf{R}$

Let $a, b \in \mathbf{R}$ and $a<b$. Then the set of real numbers $\{y: a<y<b\}$ is called an open interval and is denoted by $(a, b)$. All the points between $a$ and $b$ belong to the open interval $(a, b)$ but $a, b$ themselves do not belong to this interval.

The interval which contains the end points also is called closed interval and is denoted by $[a, b]$. Thus

$[a, b]=\{x: a \leq x \leq b\}$

We can also have intervals closed at one end and open at the other, i.e.,

$[a, b]=\{x: a \leq x<b\}$ is an open interval from $a$ to $b$, including $a$ but excluding $b$.

$(a, b]=\{x: a<x \leq b\}$ is an open interval from $a$ to $b$ including $b$ but excluding $a$.

These notations provide an alternative way of designating the subsets of set of real numbers. For example, if $A=(-3,5)$ and $B=[-7,9]$, then $A \subset B$. The set $[0, \infty)$ defines the set of non-negative real numbers, while set $(-\infty, 0)$ defines the set of negative real numbers. The set $(-\infty, \infty)$ describes the set of real numbers in relation to a line extending from $-\infty$ to $\infty$.

On real number line, various types of intervals described above as subsets of $\mathbf{R}$, are shown in the Fig 1.1.

Here, we note that an interval contains infinitely many points.

For example, the set $\{x: x \in \mathbf{R},-5<x \leq 7\}$, written in set-builder form, can be written in the form of interval as $(-5,7]$ and the interval $[-3,5)$ can be written in setbuilder form as $\{x:-3 \leq x<5\}$.

The number $(b-a)$ is called the length of any of the intervals $(a, b),[a, b]$, $[a, b)$ or $(a, b]$.

### 1.7 Universal Set

Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “*Universal Set*”. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set $\mathbf{R}$ of real numbers. For another example, in human population studies, the universal set consists of all the people in the world.

### 1.8 Venn Diagrams

Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles.

In Venn diagrams, the elements of the sets are written in their respective circles (Figs 1.2 and 1.3)

**Illustration 1** In Fig 1.2, $U=\{1,2,3, \ldots, 10\}$ is the universal set of which $A=\{2,4,6,8,10\}$ is a subset.

**Illustration 2** In Fig 1.3, $U=\{1,2,3, \ldots, 10\}$ is the universal set of which

$A=\{2,4,6,8,10\}$ and $B=\{4,6\}$ are subsets, and also $B \subset A$.

The reader will see an extensive use of the Venn diagrams when we discuss the union, intersection and difference of sets.

### 1.9 Operations on Sets

In earlier classes, we have learnt how to perform the operations of addition, subtraction, multiplication and division on numbers. Each one of these operations was performed on a pair of numbers to get another number. For example, when we perform the operation of addition on the pair of numbers 5 and 13, we get the number 18. Again, performing the operation of multiplication on the pair of numbers 5 and 13 , we get 65 .

Similarly, there are some operations which when performed on two sets give rise to another set. We will now define certain operations on sets and examine their properties. Henceforth, we will refer all our sets as subsets of some universal set.

#### 1.9.1 *Union of sets*

Let A and B be any two sets. The union of A and B is the set which consists of all the elements of $A$ and all the elements of $B$, the common elements being taken only once. The symbol ’ $\cup$ ’ is used to denote the union. Symbolically, we write $A \cup B$ and usually read as ‘A union $B$ ‘.

**Definition 5** The union of two sets $A$ and $B$ is the set $C$ which consists of all those elements which are either in A or in B (including those which are in both). In symbols, we write. $A \cup B=\{x: x \in A$ or $x \in B\}$

The union of two sets can be represented by a Venn diagram as shown in Fig 1.4.

The shaded portion in Fig 1.4 represents $A \cup B$.

Some Properties of the Operation of Union

(i) $A \cup B=B \cup A$ (Commutative law)

(ii) $(A \cup B) \cup C=A \cup(B \cup C)$ (Associative law )

(iii) $A \cup \phi=A \quad$ (Law of identity element, $\phi$ is the identity of $\cup$ )

(iv) $A \cup A=A \quad$ (Idempotent law)

(v) $U \cup A=U \quad$ (Law of $\cup$)

#### 1.9.2 *Intersection of sets*

The intersection of sets A and B is the set of all elements which are common to both A and B. The symbol ’ $\cap$ ’ is used to denote the intersection. The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write $A \cap B=\{x: x \in A$ and $x \in B\}$.

**Definition 6** The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write

$A \cap B=\{x: x \in A$ and $x \in B\}$

The shaded portion in Fig 1.5 indicates the intersection of $A$ and $B$.

If $A$ and $B$ are two sets such that $A \cap B=\phi$, then $A$ and $B$ are called disjoint sets.

For example, let $A=\{2,4,6,8\}$ and

$B=\{1,3,5,7\}$. Then $A$ and $B$ are disjoint sets, because there are no elements which are common to A and B. The disjoint sets can be represented by means of Venn diagram as shown in the Fig 1.6 In the above diagram, $A$ and $B$ are disjoint sets.

Some Properties of Operation of Intersection

(i) $A \cap B=B \cap A \quad$ (Commutative law).

(ii) $(A \cap B) \cap C=A \cap(B \cap C) \quad$ (Associative law).

(iii) $\phi \cap A=\phi, \cup \cap A=A \quad$ (Law of $\phi$ and $\cup$ ).

(iv) $A \cap A=A \quad$ (Idempotent law) (v) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$ (Distributive law )i. e., $\cap$ distributes over $\cup$

This can be seen easily from the following Venn diagrams [Figs 1.7 (i) to (v)].

#### 1.9.3 *Difference of sets*

The difference of the sets A and B in this order is the set of elements which belong to A but not to B. Symbolically, we write A - B and read as “A minus B”.

### 1.10 Complement of a Set

Let $\cup$ be the universal set which consists of all prime numbers and $A$ be the subset of $\cup$ which consists of all those prime numbers that are not divisors of 42 . Thus, $A=\{x: x \in \cup$ and $x$ is not a divisor of 42 $\}$. We see that $2 \in U$ but $2 \notin A$, because 2 is divisor of 42 . Similarly, $3 \in \cup$ but $3 \notin A$, and $7 \in \cup$ but $7 \notin A$. Now 2,3 and 7 are the only elements of $\cup$ which do not belong to $A$. The set of these three prime numbers, i.e., the set $\{2,3,7\}$ is called the Complement of $A$ with respect to $\cup$, and is denoted by $A^{\prime}$. So we have $A^{\prime}=\{2,3,7\}$. Thus, we see that

$$ A^{\prime}=\{x: x \in U \text { and } x \notin A\} \text { This leads to the following definition. } $$

**Definition 7** Let $\cup$ be the universal set and $A$ a subset of $\cup$. Then the complement of $A$ is the set of all elements of $\cup$ which are not the elements of A. Symbolically, we write $A^{\prime}$ to denote the complement of $A$ with respect to $\cup$. Thus,

$$ A^{\prime}=\{x: x \in U \text { and } x \notin A\} \text {. Obviously } A^{\prime}=U-A $$

We note that the complement of a set A can be looked upon, alternatively, as the difference between a universal set $\cup$ and the set $A$.

**Some Properties of Complement Sets**

**1.** Complement laws:

(i) $A \cup A^{\prime}=U$

(ii) $A \cap A^{\prime}=\phi$

**2.** De Morgan’s law:

(i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$

(ii) $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$

**3.** Law of double complementation: $(A^{\prime})^{\prime}=A$

**4.** Laws of empty set and universal set $\phi^{\prime}= \cup$ and $ \cup^{\prime}=\phi$.

These laws can be verified by using Venn diagrams.

### Summary

This chapter deals with some basic definitions and operations involving sets. These are summarised below:

A set is a well-defined collection of objects.

A set which does not contain any element is called empty set.

A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set.

Two sets A and B are said to be equal if they have exactly the same elements.

A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of $\mathbf{R}$.

The union of two sets A and B is the set of all those elements which are either in $A$ or in $B$.

The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to $A$ but not to $B$.

The complement of a subset $A$ of universal set $\cup$ is the set of all elements of $\cup$ which are not the elements of $A$.

For any two sets $A$ and $B,(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ and $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$

### Historical Note

The modern theory of sets is considered to have been originated largely by the German mathematician Georg Cantor (1845-1918). His papers on set theory appeared sometimes during 1874 to 1897 . His study of set theory came when he was studying trigonometric series of the form $a_1 \sin x+a_2 \sin 2 x+a_3 \sin 3 x+\ldots$ He published in a paper in 1874 that the set of real numbers could not be put into one-to-one correspondence wih the integers. From 1879 onwards, he publishd several papers showing various properties of abstract sets.

Cantor’s work was well received by another famous mathematician Richard Dedekind (1831-1916). But Kronecker (1810-1893) castigated him for regarding infinite set the same way as finite sets. Another German mathematician Gottlob Frege, at the turn of the century, presented the set theory as principles of logic. Till then the entire set theory was based on the assumption of the existence of the set of all sets. It was the famous Englih Philosopher Bertand Russell (18721970 ) who showed in 1902 that the assumption of existence of a set of all sets leads to a contradiction. This led to the famous Russell’s Paradox. Paul R.Halmos writes about it in his book ‘Naïve Set Theory’ that “nothing contains everything”.

The Russell’s Paradox was not the only one which arose in set theory. Many paradoxes were produced later by several mathematicians and logicians. As a consequence of all these paradoxes, the first axiomatisation of set theory was published in 1908 by Ernst Zermelo. Another one was proposed by Abraham Fraenkel in 1922. John Von Neumann in 1925 introduced explicitly the axiom of regularity. Later in 1937 Paul Bernays gave a set of more satisfactory axiomatisation. A modification of these axioms was done by Kurt Gödel in his monograph in 1940. This was known as Von Neumann-Bernays (VNB) or GödelBernays (GB) set theory.

Despite all these difficulties, Cantor’s set theory is used in present day mathematics. In fact, these days most of the concepts and results in mathematics are expressed in the set theoretic language.