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}$