# Mathematical Notations

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: $n!=1\cdot 2 \cdot 3 \ldots (n-1) n /; n \in \mathbb{N}$