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:

• Odd natural numbers less than 10 , i.e., \(1,3,5,7,9\)
• The rivers of India
• The vowels in the English alphabet, namely, \(a, e, i, o, u\)
• Various kinds of triangles
• Prime factors of 210 , namely, \(2,3,5\) and 7
• 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 collection.

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

\(N\) : the set of all natural numbers
\(Z\) : the set of all integers
\(Q\) : the set of all rational numbers
\(R\) : the set of real numbers
\(Z ^{+}\): the set of positive integers
\(Q ^{+}\): the set of positive rational numbers, and
\(R ^{+}\): 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 :

• 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 \(\mathrm{A}\), we say that ” \(a\) belongs to \(\mathrm{A} “\) the Greek symbol \(\in\) (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write \(a \in \mathrm{A}\). If ‘ \(b\) ‘ is not an element of a set \(\mathrm{A}\), we write \(b \notin \mathrm{A}\) and read ” \(b\) does not belong to \(\mathrm{A}\) “.

Sets in the Mathematics world

The symbols for the special sets given below will be referred to throughout this chapter.

• \(\mathbf{N}\): the set of all natural numbers [Natural numbers come under real numbers and include the positive integers {1, 2, 3, 4, 5, 6, 7, 8…..} and so on.]
• \(\mathbf{Z}\): the set of all integers [Integers come in three types: Zero (0),Positive Integers (Natural numbers), and Negative Integers, Excample: \(\mathbf{Z}=\{\ldots \ldots-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8, \ldots \ldots\}\)]
• \(\mathbf{Q}\): the set of all rational numbers [Rational numbers are in the form of \(p / q\), where \(p\) and \(q\) can be any integer and \(q \neq 0\). This means that rational numbers include natural numbers, whole numbers, integers, fractions of integers, and decimals. For example: 3/4, 0.125, 0.333…, 1.1, etc. An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point. Some of the common irrational numbers are: \(\mathrm{Pi}(\pi)=3.141592 \ldots\)Euler’s Number \((\mathrm{e})=2.7182818284590452 \ldots \ldots\);\(\sqrt{2}=1.414213 \ldots\)]
• \(\mathbf{R}\): the set of real numbers [The set of real numbers consists of the set of rational numbers and the set of irrational numbers.]
• \(\mathbf{Z}^{+}\): the set of positive integers [The set of positive integers is represented as \(Z^{+}\). It implies all the integers greater than 0. In the roster form, it is represented as \(\{1,2,3,4\), \(5,6,7, \ldots .\}\). The number of positive integers is infinite as there is no end to the counting. Therefore, the set of positive integers is an infinite set as the number of elements is endless.]
• \(\mathbf{Q}^{+}\): the set of positive rational numbers [Positive rational numbers refer to rational numbers when their numerators and denominators are both positive or both negative. Examples of positive rational numbers are \(3 / 8,9 / 10,-34 /-40\), etc.]
• \(\mathbf{R}^{+}\): the set of positive real numbers. [the set of positive real numbers, is the subset of those real numbers that are greater than zero]

There are two methods of representing a set

• Roster or tabular form
• Set builder form

In roster form, all the elements of a set are listed, and the elements are 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, . . .}. The dots tell us that the list of odd numbers continues 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.]

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

\(\mathrm{V}=\{x: x\) is a vowel in English alphabet \(\}\)

The above description of the set \(\mathrm{V}\) 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
\(\mathrm{A}=\{x: x\) 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 \(\mathrm{A}, \mathrm{B}\), \(\mathrm{C}\), respectively, then \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) can also be represented in set-builder form as follows:
\(\mathrm{A}=\{x: x\) is a natural number which divides 42\(\}\)
\(\mathrm{B}=\{y: y\) is a vowel in the English alphabet \(\}\)
\(\mathrm{C}=\{z: z\) 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
\((x-1)(x+2)=0 \text {, i. e., } \quad x=1,-2\); Therefore, the solution set of the given equation can be written in roster form as {1, – 2}.

Example 2: \(\text { Write the set }\left\{x: x \text { is a positive integer and } x^{2}<40\right\} \text { in the roster form. }\)

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 \(A=\{1,4,9,16,25, \ldots\}\) in set-builder form.

Solution: We may write the set \(\mathrm{A}\) as
\(\mathrm{A}=\{x: x\) is the square of a natural number \(\}\)
Alternatively, we can write
\(\mathrm{A}=\left\{x: x=n^{2} \text {, where } n \in \mathbf{N}\right\}\)

Example 4: Write the set \(\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\right\}\) 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
\(\left\{x: x=\frac{n}{n+1} \text {, where } n \text { is a natural number and } 1 \leq n \leq 6\right\}\)

Example 5: Match each of the set on the left described in the roster form with the same set on the right described in the set-builder form :
(i) \(\{\mathrm{P}, \mathrm{R}, \mathrm{I}, \mathrm{N}, \mathrm{C}, \mathrm{A}, \mathrm{L}\}\) (a) \(\{x: x\) is a positive integer and is a divisor of 18\(\}\)
(ii) \(\{0\}\)  (b) \(\left\{x: x\right.\) is an integer and \(\left.x^{2}-9=0\right\}\)
(iii) \(\{1,2,3,6,9,18\}\)  (c) \(\{x: x\) is an integer and \(x+1=1\}\)
(iv) \(\{3,-3\}\)  (d) \(\{x: x\) 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).