Mathematical Notations Cheatsheet

This is a short list of some math notations that we discussed during class that can be helpful for review and reading.

1. $\Sigma$ (Sum notation)

Sum notation is used to compress writing sum of terms if there's a pattern. It's just a different way of writing sums on paper.

Example 1:
The summation $1+2+3+...+1000$ can be written as

$$1+2+...+1000=\sum_{n=1}^{1000} \, n$$

It can be read as "the sum of $n$ from $n$ equals to 1 up to 1000"
Example 2:
The summation $2+4+8+16+...+2^{100}$ can be written as

$$2^1+2^2+2^3+2^4+...+2^{100}=\sum_{n=1}^{100}\, 2^n$$

This notation can be read as "the sum of $2^n$ from $n=1$ up to $n=100$"
Example 3:
If you have data points of heights in meters:

$$x_1=1.7, x_2=1.52, x_3=1.81, x_4=1.49, x_5=1.75$$

you can write the summation as:

$$x_1+x_2+x_3+x_4+x_5 = \sum_{i=1}^5 \, x_i$$

It can be read as "the sum of $x_i$ for $i=1$ to $i=5$."

2. $P(\cdot | \cdot)$ (conditional probability)

The bar in the probability notation is used for conditional probability when a posterior event has a priori.

You would read $P(A|B)$ as "the probability of $A$ given that $B$ happened"

3. $\infty$ (infinity symbol)

It's just a symbol to denote infinity. We would have $+\infty$ for positive infinity, and $-\infty$ for negative infinity.

4. $[0,1] \& (0,1)$ (open and closed brackets for ranges)

I describing ranges, the closed bracket "$[$" means to include the value, and the closed bracket "$($" means to exclude that value.

Example 1

• $[0, 1]$ is the set of all real numbers between 0 and 1, including 0 and 1.
• $(0, 1)$ is the set of all real numbers between 0 and 1, excluding 0 and 1.
• $[0, 1)$ is the set of all real numbers between 0 and 1, including 0 and excluding 1.
  Example 2
  When we are describing ranges up to positive infinity, for example, the notation to be used is to have open brackets:
• $[0, +\infty)$ the set of all non-negative real numbers.
• $(0, +\infty)$ the set of all positive real numbers.

5. $\mathbb{R}$ (set of real numbers)

The symbol $\mathbb{R}$ denotes the set of all real numbers. We can also write it as $(-\infty, +\infty)$.

We use $\mathbb{N}$ to denote the set of natural numbers: $\{1, 2, 3, ...\}$,
$\mathbb{Z}$ for integers $\{...,-2,-1,0,1,2,...\}$,
$\mathbb{Z}^+$ for positive integers $\{1, 2, 3, ...\}=\mathbb{N}$,
$\mathbb{Z}^-$ for negative integers $\{-1, -2, -3, ...\}$, and
$\mathbb{Z}_{\ge 0}$ for non-negative integers $\{0, 1, 2, 3, ...\}$.

6. $\in$ (in symbol)

The symbol $\in$ means "is an element of" or "belongs to" a set.

Example

1. $\text{Head} \in \{\text{Head}, \text{Tail}\}$ (Head is an element of the set {Head, Tail})
2. $2 \in \{1, 2, 3, 4\}$ (2 belongs to the set {1, 2, 3, 4})

7. $\cup, \cap$ (union and intersection)

The symbols $\cup$ and $\cap$ are used when dealing with sets.

• $\cup$ means union: it represents all elements that are in either set (or both).
• $\cap$ means intersection: it represents only the elements that are in both sets.

Example
If $A = {1, 2, 3}$ and $B = {3, 4, 5}$, then:
$A \cup B = {1, 2, 3, 4, 5}$ (all unique elements in either set)
$A \cap B = {3}$ (only what they share)

8. $\bar{x}$ (average notation)

The notation $\bar{x}$ is used to represent the mean (average) of a set of numbers.

Example
Given data points: $x_1=3, x_2=5, x_3=7$, the average is:

$$\bar{x} = \frac{1}{3}\sum_{i=1}^3 \, x_i = \frac{1}{3}(x_1+x_2+x_3)=\frac{1}{3}(3+5+7)=5$$

This is often used in statistics to denote the sample mean.

9. $\mu, \sigma^2$ (mean and variance of normal distribution)

In probability and statistics, especially with the normal distribution, the symbols $\mu$ and $\sigma^2$ are used:

$\mu$ (mu) denotes the mean (or expected value).
$\sigma^2$ (sigma squared) denotes the variance, which measures how spread out the values are.

Example
In a normal distribution $X \sim \mathcal{N}(\mu, \sigma^2)$ ($X$ is sampled from a normal distribution with mean $\mu$ and variance $\sigma^2$):

• $\mu = 170$ cm could represent average height.
• $\sigma^2 = 25$ means the variance is 25, so standard deviation $\sigma = 5$.

10. $\to$ in $f:A \to B$ vs. $f(x)\to a$ (arrow for maps and approaches)

The symbol $\to$ can mean different things depending on context:

1. In functions:
   • $f: A \to B$ means “$f$ is a function from set $A$ to set $B$”.
   • For example, $f: \mathbb{R} \to \mathbb{R}$ means the function takes real numbers as inputs and outputs real numbers.
2. In limits:
   • $f(x) \to a$ as $x \to b$ means “as $x$ approaches $b$, $f(x)$ approaches $a$”.
   • For example: as $x \to +\infty$, $\frac{1}{x} \to 0$ (as $x$ goes to $+\infty$, $\frac{1}{x}$ goes to $0$).

So, in short:

• $f: A \to B$: function definition (domain → codomain)
• $x \to a$: limit or convergence behavior (approaching a value)