sets

Chapter 1: Sets

1.1 Introduction

The concept of sets is fundamental in present-day mathematics.
It's used in almost every branch of mathematics.
Sets are used to define relations and functions.
Knowledge of sets is required for studying geometry, sequences, probability, etc.
The theory of sets was developed by German mathematician Georg Cantor (1845-1918).
Cantor first encountered sets while working on problems on trigonometric series.
This chapter discusses 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, we also come across collections, for example, of natural numbers, points, prime numbers, etc.
Examples of collections:
- Odd natural numbers less than 10: 1, 3, 5, 7, 9
- The rivers of India
- The vowels in the English alphabet: a, e, i, o, u
- Various kinds of triangles
- Prime factors of 210: 2, 3, 5, and 7
- The solution of the equation $x^2 – 5x + 6 = 0$ : 2 and 3
Each of the above examples 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, the river Nile does not belong to the collection of rivers of India.
- The river Ganga does belong to this collection.
Examples of sets used particularly in mathematics:
- $\mathbb{N}$ : the set of all natural numbers
- $\mathbb{Z}$ : the set of all integers
- $\mathbb{Q}$ : the set of all rational numbers
- $\mathbb{R}$ : the set of real numbers
- $\mathbb{Z^+}$ : the set of positive integers
- $\mathbb{Q^+}$ : the set of positive rational numbers
- $\mathbb{R^+}$ : the set of positive real numbers
The symbols for the special sets given above will be referred to throughout this text.
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.
A set is a well-defined collection of objects.
The following points may be noted:
- Objects, elements, and members of a set are synonymous terms.
- Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
- 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 ∈ (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:
- Roster or tabular form
- Set-builder form

(i) Roster Form

In roster form, all the elements of a set are listed, the elements being separated by commas and 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:
- 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}.
- The set of all vowels in the English alphabet is {a, e, i, o, u}.
- The set of odd natural numbers is represented by {1, 3, 5, . . .}. 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) Set-Builder Form

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 possesses this property.
Denoting this set by V, we write
$V = {x : x \text{ is a vowel in English alphabet}}$
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 \text{ 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 \text{ is a natural number which divides 42}} $

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

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

Example 1: Write the solution set of the equation $x^2 + x – 2 = 0$ in roster form.
Solution:
The given equation can be written as $(x – 1) (x + 2) = 0$ , i. e., $x = 1, – 2$ . Therefore, the solution set of the given equation can be written in roster form as ${1, – 2}$ .
Example 2: Write the set {x : x \text{ is a positive integer and } x^2 < 40} in the roster form.
Solution: The required numbers are 1, 2, 3, 4, 5, 6. So, the given set in the roster form is ${1, 2, 3, 4, 5, 6}$ .
Example 3: Write the set $A = {1, 4, 9, 16, 25, . . .}$ in set-builder form.
Solution: We may write the set A as
$A = {x : x \text{ is the square of a natural number}}$
Alternatively, we can write $A = {x : x = n^2, \text{ where } n \in \mathbb{N}}$
Example 4: Write the set ${\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}}$ in the set-builder form.
Solution: We see that each member in the given set has the numerator one less than the denominator. Also, the numerator begins from 1 and does not exceed 6. Hence, in the set-builder form, the given set is
${x : x = \frac{n}{n+1}, \text{ where } n \text{ is a natural number and } 1 \le n \le 6}$
Example 5: Match each of the sets on the left described in the roster form with the same set on the right described in the set-builder form:
- (i) ${P, R, I, N, C, A, L}$ (a) ${x : x \text{ is a positive integer and is a divisor of 18}}$
- (ii) ${ 0 }$ (b) ${x : x \text{ is an integer and } x^2 – 9 = 0}$
- (iii) ${1, 2, 3, 6, 9, 18}$ (c) ${x : x \text{ is an integer and } x + 1= 1}$
- (iv) ${3, –3}$ (d) ${x : x \text{ is a letter of the word PRINCIPAL}}$
Solution:
- Since in (d), there are 9 letters in the word PRINCIPAL and two letters P and I are repeated, so (i) matches (d).
- Similarly, (ii) matches (c) as $x + 1 = 1$ implies $x = 0$ .
- Also, 1, 2, 3, 6, 9, 18 are all divisors of 18, and so (iii) matches (a).
- Finally, $x^2 – 9 = 0$ implies $x = 3, –3$ , and so (iv) matches (b).

1.3 The Empty Set

Consider the set A = ${ x : x \text{ is a student of Class XI presently studying in a school }}$
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:
$B = { x : x \text{ is a student presently studying in both Classes X and XI }}$
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 φ or ${}$ .
Examples of empty sets:
- (i) Let A = {x : 1 < x < 2, x \text{ is a natural number}}. Then A is the empty set because there is no natural number between 1 and 2.
- (ii) $B = {x : x^2 – 2 = 0 \text{ and } x \text{ is rational number}}$ . Then B is the empty set because the equation $x^2 – 2 = 0$ is not satisfied by any rational value of x.
- (iii) $C = {x : x \text{ is an even prime number greater than 2}}$ . Then C is the empty set because 2 is the only even prime number.
- (iv) $D = { x : x^2 = 4, x \text{ is odd }}$ . 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 $A = {1, 2, 3, 4, 5}$ , $B = {a, b, c, d, e, g}$ , and $C = {\text{men living presently in different parts of the world}}$
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 the 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 a 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 an 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 $n(C) = \text{some finite number}$ .
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, ${1, 2, 3 . . .}$ is the set of natural numbers, ${1, 3, 5, 7, . . .}$ is the set of odd natural numbers, ${. . .,–3, –2, –1, 0,1, 2 ,3, . . .}$ 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.
Example 6: State which of the following sets are finite or infinite:
- (i) ${x : x \in \mathbb{N} \text{ and } (x – 1) (x –2) = 0}$
- (ii) ${x : x \in \mathbb{N} \text{ and } x^2 = 4}$
- (iii) ${x : x \in \mathbb{N} \text{ and } 2x –1 = 0}$
- (iv) ${x : x \in \mathbb{N} \text{ and } x \text{ is prime}}$
- (v) ${x : x \in \mathbb{N} \text{ and } x \text{ is odd}}$
Solution:
- (i) Given set = ${1, 2}$ . Hence, it is finite.
- (ii) Given set = ${2}$ . Hence, it is finite.
- (iii) Given set = φ. Hence, it is finite.
- (iv) The given set is the set of all prime numbers, and since the set of prime numbers is infinite, hence the given set is infinite.
- (v) Since there are an infinite number of odd numbers, hence the given set is infinite.

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 $A \ne B$ .
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 be 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.
Example 7: Find the pairs of equal sets, if any, give reasons:
$A = {0}$ , B = {x : x > 15 \text{ and } x < 5}, $C = {x : x – 5 = 0 }$ , $D = {x: x^2 = 25}$ , $E = {x : x \text{ is an integral positive root of the equation } x^2 – 2x –15 = 0}$ .
Solution:
Since $0 \in A$ and 0 does not belong to any of the sets B, C, D, and E, it follows that $A \ne B$ , $A \ne C$ , $A \ne D$ , $A \ne E$ .
Since $B = φ$ but none of the other sets are empty. Therefore, $B \ne C$ , $B \ne D$ , and $B \ne E$ .
Also, $C = {5}$ but $–5 \in D$ , hence $C \ne D$ . Since $E = {5}$ , $C = E$ .
Further, $D = {–5, 5}$ and $E = {5}$ , we find that, $D \ne E$ .
Thus, the only pair of equal sets is C and E.
Example 8: Which of the following pairs of sets are equal? Justify your answer.
- (i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”.
- (ii) $A = {n : n \in \mathbb{Z} \text{ and } n^2 \le 4}$ and $B = {x : x \in \mathbb{R} \text{ and } x^2 – 3x + 2 = 0}$ .
Solution:
- (i) We have, $X = {A, L, L, O, Y}$ , $B = {L, O, Y, A, L}$ . Then X and B are equal sets as the repetition of elements in a set do not change a set. Thus, $X = {A, L, O, Y} = B$
- (ii) $A = {–2, –1, 0, 1, 2}$ , $B = {1, 2}$ . Since $0 \in A$ and $0 \notin B$ , A and B are not equal sets.

1.6 Subsets

Consider the sets:
$X = \text{set of all students in your school}$ , $Y = \text{set of all students in your class}$ .
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 a 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 definition 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
$A \subset B \text{ and } B \subset A \Leftrightarrow A = B$
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 φ has no elements, we agree to say that φ is a subset of every set.
We now consider some examples:
- (i) The set Q of rational numbers is a subset of the set R of real numbers, and we write $Q \subset R$ .
- (ii) If A is the set of all divisors of 56 and B is 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 $B = {x : x \text{ is an odd natural number less than 6}}$ . 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 $A \ne B$ , then A is called a proper subset of B, and B is called the 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.
Example 9: Consider the sets φ, $A = { 1, 3 }$ , $B = {1, 5, 9}$ , $C = {1, 3, 5, 7, 9}$ . Insert the symbol $\subset$ or $\not\subset$ between each of the following pair of sets:
- (i) φ…B
- (ii) A…B
- (iii) A…C
- (iv) B…C
Solution:
- (i) φ $\subset$ B as φ is a subset of every set.
- (ii) A $\not\subset$ B as $3 \in A$ and $3 \notin B$
- (iii) A $\subset$ C as 1, 3 $\in$ A also belongs to C
- (iv) B $\subset$ C as each element of B is also an element of C.
Example 10: Let $A = { a, e, i, o, u}$ and $B = { a, b, c, d}$ . Is A a subset of B? No. (Why?). Is B a subset of A? No. (Why?)
Example 11: Let A, B, and C be three sets. If $A \in B$ and $B \subset C$ , is it true that $A \subset C$ ?. If not, give an example.
Solution:
- No. Let $A = {1}$ , $B = {{1}, 2}$ , and $C = {{1}, 2, 3}$ . Here $A \in B$ as $A = {1}$ and $B \subset C$ . But $A \not\subset C$ as $1 \in A$ and $1 \notin C$ .
  Note that an element of a set can never be a subset of itself.

1.6.1 Subsets of the set of real numbers

As noted in Section 1.6, there are many important subsets of R. We give below the names of some of these subsets.
- The set of natural numbers $\mathbb{N} = {1, 2, 3, 4, 5, . . .}$
- The set of integers $\mathbb{Z} = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}$
- The set of rational numbers $\mathbb{Q} = { x : x = \frac{p}{q} , p, q \in \mathbb{Z} \text{ and } q \ne 0}$
Which is read “ 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 Q include –5 (which can be expressed as $\frac{-5}{1}$ ), $\frac{7}{5}$ , $1\frac{3}{2}$ (which can be expressed as $\frac{7}{2}$ ), and $\frac{-11}{3}$ .
The set of irrational numbers, denoted by T, is composed of all other real numbers.
Thus $\mathbb{T} = {x : x \in \mathbb{R} \text{ and } x \notin \mathbb{Q}}$ , i.e., all real numbers that are not rational.
Members of T include $\sqrt{2}$ , $\sqrt{5}$ , and $\pi$ .
Some of the obvious relations among these subsets are:

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

1.6.2 Intervals as subsets of R

Let $a, b \in \mathbb{R}$ 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 the closed interval and is denoted by $[ a, b ]$ . Thus $[ a, b ] = {x : a \le x \le b}$
We can also have intervals closed at one end and open at the other, i.e.,
[ a, b ) = {x : a \le x < b} is an open interval from a to b, including a but excluding b.
( a, b ] = { x : a < x \le 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 the 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 the real number line, various types of intervals described above as subsets of 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 \mathbb{R}, –5 < x \le 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 set-builder form as {x : –3 \le 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 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.
Illustration 1: In Fig 1.2, $U = {1,2,3, …, 10}$ is the universal set of which $A = {2,4,6,8,10}$ is a subset.
Illustration 2: In Fig 1.3, $U = {1,2,3, …, 10}$ 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 learned 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 to 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 ‘∪’ is used to denote the union. Symbolically, we write $A ∪ B$ and usually read as ‘A union B’.
Example 12: Let $A = { 2, 4, 6, 8}$ and $B = { 6, 8, 10, 12}$ . Find $A ∪ B$ .
Solution: We have $A ∪ B = { 2, 4, 6, 8, 10, 12}$ . Note that the common elements 6 and 8 have been taken only once while writing $A ∪ B$ .
Example 13: Let $$