0.2.1 Symbols, sets and function notation

A formula is a sentence

Every symbol in a formula stands for a word or two. Here is the average of $n$ numbers:

\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}

Read it aloud: "x-bar equals x-one plus x-two, and so on up to x-n, all divided by n." Once you can say a formula, you can write it in code.

Symbol	Say it	It means
$n$	"n"	how many values there are
$x_1$	"x one" or "x sub one"	the first value
$x_i$	"x i"	the value in position $i$
$\cdots$	"and so on"	the pattern carries on
$\bar{x}$	"x bar"	the average of the $x$ values

Subscripts are positions

A subscript is a small number or letter written low and to the right. It labels a position in a list: $x_1$ is the first value, $x_2$ the second, and $x_i$ is "the $i$ -th one", whichever $i$ you pick.

Rule: Subscripts

\begin{gathered} x_1, x_2, \ldots, x_n \\ \updownarrow \\ \texttt{x[0]}, \texttt{x[1]}, \ldots, \texttt{x[n-1]} \end{gathered}

Maths usually counts from 1. Python counts from 0. So $x_1$ is x[0], and $x_i$ is x[i - 1].

x_1 = 4

⌘/Ctrl + Enter runs the cell

Sets and membership

A set is a collection of different objects, written inside curly brackets: $A = \{2, 3, 5, 7\}$ . The objects are its elements. Order does not matter and repeats do not count, so $\{1, 2, 2, 3\}$ is the same set as $\{3, 2, 1\}$ .

Rule: Membership

x \in A \qquad x \notin A

$x \in A$ says " $x$ is an element of $A$ ". $x \notin A$ says it is not. With $A = \{2, 3, 5, 7\}$ : $3 \in A$ and $4 \notin A$ .

A few sets have their own letters, and they appear everywhere:

Symbol	Name	What is in it
$\mathbb{N}$	natural numbers	$1, 2, 3, \ldots$ (some books start at 0)
$\mathbb{Z}$	integers	$\ldots, -2, -1, 0, 1, 2, \ldots$
$\mathbb{R}$	real numbers	every point on the number line, such as $-1.5$ , $0$ , $\sqrt{2}$
$[a, b]$	closed interval	every real number from $a$ to $b$ , ends included
$(a, b)$	open interval	every real number strictly between $a$ and $b$

\in

⌘/Ctrl + Enter runs the cell

Set-builder notation: "such that"

Listing every element only works for small sets. Set-builder notation describes a set by a rule instead:

\begin{gathered} \{x \in \mathbb{Z} : 1 \le x \le 10,\ x \text{ is even}\} \\ = \{2, 4, 6, 8, 10\} \end{gathered}

Read the colon as "such that": "the set of integers $x$ such that $x$ is between 1 and 10 and $x$ is even". Some books write a vertical bar instead of the colon. It means the same thing.

Rule: Set-builder notation

\{x \in A : \text{condition}\}

Everything in $A$ that passes the condition. The part before the colon says where to look. The part after says what to keep.

Try it A Python set comprehension reads almost word for word like set-builder notation. range(1, 11) gives the integers from 1 to 10, because a range stops just before its second number. x % 2 == 0 tests "x is even".

⌘/Ctrl + Enter runs the cell

"For all" and "there exists"

Two symbols turn a condition into a statement that is true or false about a whole set.

Rule: Quantifiers

\begin{gathered} \forall x \in A,\ P(x) \\ \exists\, x \in A : P(x) \end{gathered}

$\forall$ reads "for all". The statement is true when every element of $A$ passes the test $P$ . $\exists$ reads "there exists". It is true when at least one element passes.

With $A = \{2, 3, 5, 7\}$ :

$\forall x \in A,\ x > 1$ is true. Every element is bigger than 1.
$\exists\, x \in A : x > 6$ is true, because of 7.
$\forall x \in A,\ x \text{ is odd}$ is false. The single element 2 is enough to break it.

\forall

⌘/Ctrl + Enter runs the cell

You will also read the word let. "Let $n$ be the number of values" gives a name to something, like n = len(x) in Python. "Let $r > 0$ " means "assume $r$ is positive from here on".

Function notation

A function is a rule that turns each input into exactly one output. The formula $f(x) = x^2 + 1$ names the rule $f$ and says what it does to an input $x$ : square it, then add 1.

Rule: Function notation

\begin{gathered} f(x) = x^2 + 1 \\ f(3) = 3^2 + 1 = 10 \end{gathered}

$f(3)$ means "the output of $f$ when the input is 3". It is not $f$ times 3. To evaluate, replace every $x$ in the formula with the input.

The letter inside the brackets is only a placeholder. $f(t) = t^2 + 1$ is the same function. The input can be an expression too: $f(a + 1) = (a + 1)^2 + 1$ . Keep the brackets when you substitute.

You will also see $f: \mathbb{R} \to \mathbb{R}$ . Read it as " $f$ takes a real number and gives back a real number". The set before the arrow holds the allowed inputs; outputs land in the set after it. Lesson 1.1.1 returns to this.

f(3)

⌘/Ctrl + Enter runs the cell

Common mistakes

f(a + 1)

⌘/Ctrl + Enter runs the cell

Check yourself

Let

A = \{2, 4, 6, 9\}

. Which statement is true?

\forall x \in A,\ x \text{ is even}

\exists\, x \in A : x > 8

\forall x \in A,\ x < 9

\exists\, x \in A : x < 0

Check yourself

In Python, x = [4, 8, 15, 16]. Which maths symbol does x[2] stand for?

x_2 = 8

x^2

x_3 = 15

Symbols and functions

0 of 7 correct

1.
The values $x_1, x_2, x_3, x_4$ are 4, 8, 15, 16. Evaluate $x_2 + x_4$
2.
How many elements does $\{x \in \mathbb{Z} : 1 \le x \le 20,\ x \text{ is a multiple of } 3\}$ have?
3.
$A = \{2, 3, 5, 7\}$ . How many elements $x \in A$ satisfy $x > 2$ ?
4.
If $f(x) = x^2 + 1$ , find $f(5)$
5.
If $f(x) = 2x - 5$ , find $f(0)$
6.
If $g(t) = t^2$ , find $g(-4)$
7.
If $f(x) = 3x + 2$ , write $f(a + 1)$ in its simplest form

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

x_i

Symbols, sets and function notation

A formula is a sentence

Subscripts are positions

Rule: Subscripts

Sets and membership

Rule: Membership

Set-builder notation: "such that"

Rule: Set-builder notation

"For all" and "there exists"

Rule: Quantifiers

Function notation

Rule: Function notation

Common mistakes

Symbols and functions

Summary card

Free further reading