# Mathematical Notations

List of the most used mathematical notations that are used in expressions and formulas.

## Essential Notations

$$\sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n$$

Summation notation or Sigma notation

$$\prod_{i=1}^{n} a_i = a_1 \cdot a_2 \cdot \cdots \cdot a_n$$

Product notation or Pi notation

$$\int_a^b f(x) \, dx$$ Integral notation

$$\lim_{x \to a} f(x) = L$$ Limit notation

$$\frac{d}{dx} f(x) = f'(x)$$ Derivative notation [Leibniz's notation]

$$\frac{\partial}{\partial x} f(x, y)$$ Partial derivative notation

$$\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$$ Vector notation

$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$ Matrix notation

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ Binomial coefficient notation

$$A = \{x \in \mathbb{R} : x^2 < 4\}$$ Set notation

## Conditionals and Operators

• $r /; c$ : Relation $r$ holds under the condition $c$ .
• $a=b$ : The expression $a$ is mathematically identical to $b$ .
• $a \ne b$ : The expression a is mathematically different from $b$ .
• $x > y$ : The quantity $x$ is greater than quantity $y$ .
• $x \ge y$ : The quantity $x$ is greater than or equal to the quantity $y$ .
• $x < y$ : The quantity $x$ is less than quantity $y$ .
• $x \le y$ : The quantity $x$ is less than or equal to quantity $y$ .
• $P := Q$ : Statement $P$ defines statement $Q$ .
• $a \wedge b$ : a and b.
• $a \vee b$ : a or b.
• $\forall a$ : for all $a$ .
• $\exists$ : [there] exists.
• $\iff$ : If and only if.

## Sets & Domains

• $\{ a_1, a_2, \ldots, a_n \}$ : A finite set with some elements $a_1, a_2, \ldots, a_n$ .
• $\{ a_1, a_2, \ldots, a_n \ldots \}$ : An infinite set with elements $a_1, a_2, \ldots$
• $\mathrm{\{ listElement /; domainSpecification\}}$ : A sequence of elements listElement with some domainSpecifications in the set. For example, $\{ x : x=\frac{p}{q} /; p \in \mathbb{Z}, q \in \mathbb{N^+}\}$ $a \in A$ : $a$ is an element of the set A.
• $a \notin A$ : a is not an element of the set A.
• $x \in (a,b)$ : The number x lies within the specified interval $(a,b)$ .
• $x \notin (a,b)$ : The number x does not belong to the specified interval $(a,b)$.

## Standard Set Notations

• $\mathbb{N}$ : the set of natural numbers $\{0, 1, 2, \ldots \}$
• $\mathbb{N}^+$ : The set of positive natural numbers: $\{1, 2, 3, \ldots \}$
• $\mathbb{Z}$ : The set of integers $\{ 0, \pm 1, \pm 2, \ldots\}$
• $\mathbb{Q}$ : The set of rational numbers
• $\mathbb{R}$ : The set of real numbers
• $\mathbb{C}$ : The set of complex numbers
• $\mathbb{P}$ : The set of prime numbers.
• $\{ \}$ : The empty set.
• $\{ A \otimes B \}$ : The ordered set of sets $A$ and $B$ .
• $n!$ : Factorial of n: $n!=1\cdot 2 \cdot 3 \ldots (n-1) n /; n \in \mathbb{N}$