Sets and Their Representations

1.2 Sets and Their Representations

The Idea of a Collection in Mathematics

In everyday life, we constantly talk about collections — a pack of cards, a crowd of people, a cricket team. Mathematics does the same, but with a crucial condition: the collection must be well-defined. This means that for any given object, we must be able to decide, without ambiguity, whether it belongs to the collection or not.

Consider these examples:

• 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, 7
• The solutions of $x^2 - 5x + 6 = 0$: 2 and 3

Each of these is well-defined. We can say with certainty that the river Nile does not belong to the collection of Indian rivers, while the Ganga does. Similarly, 4 is not an odd natural number less than 10, so it is excluded.

Now contrast this with "the five most renowned mathematicians of the world." What counts as "most renowned"? Different people will have different opinions. This collection is not well-defined, and therefore it is not a set in the mathematical sense.

Standard Sets Used Throughout Mathematics

Certain sets appear so often that they have their own fixed symbols. You will see these throughout the entire NCERT syllabus, so memorise them now:

Terminology and Notation

A set is a well-defined collection of objects. The objects themselves are called elements or members of the set — all three terms mean the same thing.

Sets are usually denoted by capital letters: $A, B, C, X, Y, Z$, etc. Elements are denoted by small letters: $a, b, c, x, y, z$, etc.

If $a$ is an element of set $A$, we say "$a$ belongs to $A$" and write:

The symbol $\in$ (epsilon, from the Greek alphabet) stands for "belongs to." If $b$ is not an element of $A$, we write:

and read it as "$b$ does not belong to $A$."

For example, let $V$ be the set of vowels in the English alphabet. Then $a \in V$ but $b \notin V$. Let $P$ be the set of prime factors of 30. Then $3 \in P$ but $15 \notin P$.

Two Ways to Represent a Set

There are exactly two methods for writing a set. You must be comfortable with both and able to convert between them.

1. Roster Form (Tabular Form)

In roster form, you list all the elements of the set, separated by commas, and enclose them in curly braces $\{ \}$.

Examples:

• The set of all even positive integers less than 7: $\{2, 4, 6\}$
• The set of all natural numbers which divide 42: $\{1, 2, 3, 6, 7, 14, 21, 42\}$
• The set of all vowels in English: $\{a, e, i, o, u\}$
• The set of odd natural numbers: $\{1, 3, 5, \dots\}$

Two important rules for roster form:

1. Order does not matter. The set $\{1, 2, 3\}$ is exactly the same as $\{3, 1, 2\}$. The elements are just a collection — there is no first, second, or last.

2. Elements are not repeated. Every element appears only once. For example, the set of letters forming the word "SCHOOL" is $\{S, C, H, O, L\}$, not $\{S, C, H, O, O, L\}$. The letter O appears only once because a set contains distinct objects.

2. Set-Builder Form

In set-builder form, you describe the common property that all elements of the set share — a property that no object outside the set possesses.

The general pattern is:

The braces $\{ \}$ mean "the set of all," and the colon $:$ means "such that." So $\{x : x \text{ is a vowel in English}\}$ is read as "the set of all $x$ such that $x$ is a vowel in English."

Examples:

• $V = \{x : x \text{ is a vowel in the English alphabet}\}$
• $A = \{x : x \text{ is a natural number and } 3 < x < 10\}$

For the second example, the natural numbers between 3 and 10 (excluding 3 and 10) are 4, 5, 6, 7, 8, 9. So $A = \{4, 5, 6, 7, 8, 9\}$ in roster form.

You can use any symbol for the variable — $x$, $y$, $z$, or anything else. The following three sets are identical:

• $A = \{x : x \text{ is a natural number which divides } 42\}$
• $B = \{y : y \text{ is a vowel in the English alphabet}\}$
• $C = \{z : z \text{ is an odd natural number}\}$ …