Sets and their representations

Look around you. A library has a collection of books; a cricket team is a collection of eleven players; the digits on a number lock form a collection of ten symbols. In mathematics we want a single word for "well-defined collection". That word is set. Once we agree on what a set is, we can talk about anything: numbers, points, polygons, functions, even other sets.

Definitions

A set is a well-defined collection of distinct objects. Each object in the set is called an element (or member). We write $x \in A \quad\text{if } x \text{ is an element of } A, \qquad x \notin A \quad\text{otherwise.}$

"Well-defined" is the crucial word. The collection of "tall students in your class" is not a set , different judges disagree about what counts as tall. The collection of "students taller than $170$ cm in your class" is a set , for any specific student we can check yes or no.

Two further conventions:

1. Order does not matter. $\{1,2,3\} = \{3,1,2\}$.
2. Repetition does not count. $\{1,2,2,3\}$ is the same as $\{1,2,3\}$.

Sets are usually named with capital letters $A, B, C, \dots$ and elements with lowercase letters $a, b, c, \dots$. Some sets recur so often that they have permanent names: $\mathbb{N} = \{1,2,3,\dots\},\quad \mathbb{W} = \{0,1,2,\dots\},\quad \mathbb{Z} = \{\dots,-2,-1,0,1,2,\dots\},$ $\mathbb{Q} = \left\{ \tfrac{p}{q} : p, q \in \mathbb{Z},\ q \ne 0\right\},\quad \mathbb{R} = \text{the real numbers}.$

Roster (tabular) form

In roster form we list the elements between curly braces, separated by commas. Examples:

• Vowels of English: $V = \{a, e, i, o, u\}$.
• Odd natural numbers less than $10$: $O = \{1,3,5,7,9\}$.
• Solutions of $x^2 - 5x + 6 = 0$: $S = \{2, 3\}$.

Two cautions. The set $\{1,2,3,\dots,100\}$ uses dots when the pattern is obvious, but $\{1, 4, 9, 16, \dots\}$ requires the reader to guess the rule. If there is any ambiguity, prefer set-builder form.

Set-builder form

In set-builder form we describe the elements by a property: $A = \{x : P(x)\} \quad \text{or} \quad A = \{x \mid P(x)\},$ read as "the set of all $x$ such that $P(x)$ is true." Examples:

• $V = \{x : x \text{ is a vowel in English}\}$.
• $O = \{x : x \in \mathbb{N},\ x \text{ is odd},\ x < 10\}$.
• $S = \{x \in \mathbb{R} : x^2 - 5x + 6 = 0\}$.

A set may sometimes be described in multiple equivalent ways: $\{2,3\} = \{x \in \mathbb{R} : x^2 - 5x + 6 = 0\} = \{x \in \mathbb{N} : x \text{ is prime and } x \le 3\}.$ Two such descriptions describe the same set whenever they have the same elements.

When to use which form

Roster form is faster for small finite sets. Set-builder is essential for:

• Infinite sets. $E = \{x \in \mathbb{N} : x \text{ is even}\}$ is precise; "$\{2,4,6,\dots\}$" relies on guessing.
• Sets defined by conditions. $\{x \in \mathbb{R} : x^2 < 2\} = (-\sqrt{2}, \sqrt{2})$.
• Sets with no obvious pattern. $\{n \in \mathbb{N} : n \text{ is prime}\}$ , you cannot just list and "...".

Worked examples

Example 1. Write $A = \{x \in \mathbb{Z} : -3 \le x \le 3\}$ in roster form.

We list every integer between $-3$ and $3$ (inclusive): $A = \{-3, -2, -1, 0, 1, 2, 3\}.$

Example 2. Express the set $\{1, 4, 9, 16, 25\}$ in set-builder form.

The pattern is "square of a natural number from $1$ to $5$": $\{x : x = n^2,\ n \in \mathbb{N},\ 1 \le n \le 5\}.$

Example 3. Decide whether each is a set: (i) the collection of all good cricketers in India; (ii) the collection of all rivers in India; (iii) the collection of all even prime numbers.

(i) Not a set , "good" is not well-defined. (ii) A set , given a river, we can check whether it flows in India. (iii) A set; in fact a remarkably small one: $\{2\}$.

Example 4. Write $B = \{x : x = \tfrac{n}{n+1},\ n \in \mathbb{N},\ n \le 4\}$ in roster form.

For $n=1,2,3,4$ we get $\tfrac{1}{2}, \tfrac{2}{3}, \tfrac{3}{4}, \tfrac{4}{5}$. So $B = \left\{\tfrac{1}{2}, \tfrac{2}{3}, \tfrac{3}{4}, \tfrac{4}{5}\right\}.$

Example 5 (harder). Let $A = \{x \in \mathbb{R} : x^4 - 5x^2 + 4 = 0\}$. Write $A$ in roster form.

Put $y = x^2$. Then $y^2 - 5y + 4 = 0$, so $(y-1)(y-4) = 0$ giving $y = 1$ or $y = 4$. Hence $x^2 = 1$ or $x^2 = 4$, so $x \in \{-2, -1, 1, 2\}$. Thus $A = \{-2, -1, 1, 2\}.$

Try it yourself

1. Write in roster form: $\{x \in \mathbb{N} : x \text{ divides } 12\}$.
2. Write in roster form: $\{x \in \mathbb{Z} : x^2 \le 9\}$.
3. Convert $\{0, 1, 4, 9, 16\}$ into set-builder form.
4. Decide whether each is a set: (a) tall buildings in Mumbai; (b) prime numbers less than $20$; (c) honest politicians; (d) integers divisible by $7$ between $-30$ and $30$.
5. Write in roster form: $\{x \in \mathbb{R} : x \text{ is a real root of } x^2 + 1 = 0\}$.
6. Write in set-builder form: $\{1, 8, 27, 64, 125\}$.
7. Write all elements of $\{x \in \mathbb{N} : x \le 10,\ x \text{ is a multiple of } 3\}$.
8. List the elements of $\{x \in \mathbb{Q} : x^2 = 4\}$ and of $\{x \in \mathbb{Z} : x^2 = 2\}$.
9. True or false (with reason): the set $\{1, 1, 2, 3\}$ has four elements.
10. Write $\{x \in \mathbb{N} : x = 2k - 1,\ k \in \mathbb{N},\ k \le 6\}$ in roster form.
11. Express in roster form: $\{x : x \text{ is a letter in the word ‘MATHEMATICS’}\}$.
12. Convert $\{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x + y = 4,\ 0 \le x \le 4\}$ to roster form.

Pitfalls / Tricks

• A set is not a list: $\{1,2,3\}$ and $\{3,2,1,1\}$ are the same set.
• "Well-defined" is your shield. If a property uses words like good, beautiful, interesting, it does not define a set.
• A common trap: $\{x \in \mathbb{R} : x^2 + 1 = 0\}$ is the empty set, but $\{x \in \mathbb{C} : x^2 + 1 = 0\} = \{i, -i\}$. Always state where $x$ lives.
• Insight. Whenever a set is given by a property, ask "is there any $x$ that satisfies it?" That single check catches half of all set-theory errors.