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Sets
Introduction
Sets and their Representations
Practice: Roster and Set-Builder Form
The Empty Set
Finite and Infinite Sets
Equal Sets
Practice: Empty, Finite and Equal Sets
Subsets
Universal Set
Practice: Subsets and Intervals
Venn Diagrams
Operations on Sets
Practice: Union, Intersection and Difference
Complement of a Set
Practice: Complement of a Set
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Glossary

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SetsSets and their Representations

Reading time: ~30 min

All around us we talk about collections: a pack of cards, a cricket team, the crowd at a mela. Mathematics is full of collections too — of numbers, of points, of shapes. Here are six of them:

(i) The natural numbers less than 10: 1, 3, 5, 7, 9

(ii) The rivers of India

(iii) The of the English alphabet: a, e, i, o, u

(iv) The different kinds of triangles

(v) The prime of 210: 2, 3, 5 and 7

(vi) The solutions of the equation x^2 - 5x + 6 = 0: 2 and 3

All six are well-defined: give me any object and I can decide for certain whether it to the collection. The Ganga belongs to collection (ii); the Nile does not. Compare that with "the five greatest mathematicians of all time" — ask five people and you will get five different lists, so that collection is not well-defined.

A set is a well-defined collection of objects.

Sort these collections: which are sets, and which are not?

The prime numbers less than 50
The five best cricketers of India
The students of your class taller than 160 cm
The tall students of your class
The months that have 31 days
The tastiest dishes in the school canteen
The letters of the word SCHOOL
The most difficult chapters in this book
A set (well-defined)
Not a set

Look at the pair "the tall students" and "the students taller than 160 cm". The first depends on opinion; the second needs only a measuring tape. Turning a vague description into a precise one is exactly what turns a collection into a set.

Belongs To: the Symbol ∈

A few conventions. The objects in a set are called its elements or members — the three words mean the same. Sets are usually named with letters like A, B, C, X, Y, and their elements with small letters like a, b, x, y.

If a is an element of the set A, we say "a belongs to A" and write a \in A. The symbol \in is the Greek letter epsilon. If b is not an element of A, we write b \notin A and read it as "b does not belong to A".

For the set V of vowels, a \in V but b \notin V. For the set P of prime factors of 30, 3 \in P but 15 P — 15 does divide 30, but it is not prime.

Some sets are used so often that they have their own bold letters. You will meet them all through this book:

SymbolThe set of all …
\mathbf{N}natural numbers 1, 2, 3, …
\mathbf{Z}integers
\mathbf{Q}rational numbers
\mathbf{R}real numbers
\mathbf{Z}^+, \mathbf{Q}^+, \mathbf{R}^+positive integers, positive rationals, positive reals

So -3 \notin \mathbf{N}, but -3 \mathbf{Z}.

There are two standard ways to describe a set: list its elements (the roster form), or state the rule its elements obey (the set-builder form). First, a quick round to make \in and \notin automatic.

Speed Round: In or Out?

Each card shows an element and a set. Does the element belong to the set?

Get five right to move on.

Listing It All: Roster Form

In roster form (also called tabular form) we write every element of the set, separate them with commas and close the list in curly braces \{ \}. The even positive integers less than 7, for example, form the set \{2, 4, 6\}. A few more:

(a) The natural numbers that divide 42: \{1, 2, 3, 6, 7, 14, 21, 42\}

(b) The vowels of the English alphabet: \{a, e, i, o, u\}

(c) The odd natural numbers: \{1, 3, 5, \ldots\}. The three dots say that the pattern goes on .

Tip

Two rules for roster form. Order does not matter: \{1, 3, 7, 21, 2, 6, 14, 42\} is the same set as (a). Each element is written once: the letters of the word SCHOOL form the set \{S, C, H, O, L\} — the second O adds nothing new.

So the letters of the word APPLE form the set , written in braces.

Describing by a Rule: Set-Builder Form

Listing works for small sets, but not for "all the odd numbers". In set-builder form we describe the elements by a property that every element has and nothing outside the set has. The vowels, for example, are exactly the letters that are vowels, so

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

Read it aloud as "V is the set of all x such that x is a vowel in the English alphabet". The braces say "the set of all", the letter x stands for a typical element, and the colon says "such that". After the colon comes the rule.

The rule machine. Here are three sets in roster form. For each one, build a rule: choose what kind of number x is and how big it may be, and the machine lists the set your rule describes.

Now read one the other way round: A = \{x : x \text{ is a natural number and } 3 < x < 10\} is the set of natural numbers between 3 and 10, so A = \{4, 5, 6, 7, 8, 9\}. And the three sets (a), (b), (c) of roster form can be written as \{x : x \text{ is a natural number which divides } 42\}, \{y : y \text{ is a vowel in the English alphabet}\} and \{z : z \text{ is an odd natural number}\}. The letter you use for the element — x, y or z — makes no difference.

From Rule to List

Example 1 : Write the solution set of the equation x^2 + x - 2 = 0 in roster form.

Solution : Factorise the left-hand side: x^2 + x - 2 = (x - 1)(x + 2). The product is zero when x = or x = .

So the solution set in roster form is \{1, -2\}.

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

Solution : Try the positive integers in turn. 6^2 = 36 < 40, but 7^2 = 49 > 40, so the largest element is .

The set is \{1, 2, 3, 4, 5, 6\}.

From List to Rule

Example 3 : Write the set A = \{1, 4, 9, 16, 25, \ldots\} in set-builder form.

Solution : 1, 4, 9, 16, 25 are 1^2, 2^2, 3^2, 4^2, 5^2, so every element is the of a natural number:

A = \{x : x \text{ is the square of a natural number}\}

or, in symbols, A = \{x : x = n^2, \text{ where } n \in \mathbf{N}\}.

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 set-builder form.

Solution : In each fraction the numerator is the denominator, and the numerators run from 1 up to . So the set is

\left\{ x : x = \frac{n}{n+1}, \ n \in \mathbf{N}, \ 1 \le n \le 6 \right\}

Twin Sets

Example 5 : Match each set on the left, written in roster form, with the same set on the right, written in set-builder form.

Drag a line from each set on the left to its twin on the right.

​{P, R, I, N, C, A, L}
​{0}
​{1, 2, 3, 6, 9, 18}
​{3, −3}
​{x : x is a positive integer and a divisor of 18}
​{x : x is an integer and x² − 9 = 0}
​{x : x is an integer and x + 1 = 1}
​{x : x is a letter of the word PRINCIPAL}

Solution

The word PRINCIPAL has 9 letters, but P and I appear twice, so its letter set is \{P, R, I, N, C, A, L\}: (i) matches (d). Next, x + 1 = 1 gives x = 0, so (ii) matches (c). The numbers 1, 2, 3, 6, 9, 18 are exactly the positive divisors of 18, so (iii) matches (a). Finally x^2 - 9 = 0 gives x = 3 or x = -3, so (iv) matches (b).

Sina