Notations

Here, I would like to establish terminology and some notations that I have used in the blog. I will keep updating this page, when I happen to use newer notations in future.

• $=\ :=$ derived equality.

• $:=\ :=$ defined equality

• $\blacksquare\ :=$ end of proof.

• $logx :=$ natural logarithm of $x$, to the base $e$.

• ${log_b}x :=$ logarithm of $x$, to the base $b$.

• $exp[x] :=$ exponential function $e^x$, where $e\approx 2.71828$.

• $\implies:=$ implies

• $\iff :=$ implies and implied by, i.e., if and only if.

• $\phi :=$ empty (null) set.

• $x\in A :=$ $x$ is member of set $A$, i.e., $A$ contains $x$.

• $P(x):=$ Proposition about $x$.

• $S=\{x:P(x)\}:=$ $x\in S\iff P(x)$ holds.

• $A\subseteq B:=$ $A$ is subset of $B$, i.e., $P(x)$ holds $\forall x\in B\implies P(x)$ holds $\forall x\in A$, i.e., $x\in A\implies x\in B$.

• $A=B:=$ set $A$ and $B$ describe the same set of elements, i.e., $A\subseteq B$ and $B\subseteq A$.

• $A\subset B:=$ $A$ is proper subset of $B$, i.e., $A\subseteq B$, but $A\neq B$. Also denoted by $A\subsetneq B$.

• $\left|S\right|:=$ size or cardinality of finite set $S$.

• $A\cup B:=\{ x:x\in A\ or\ x\in B \}$, where “or” is inclusive.

• $A\cap B:=\{ x:x\in A\ and\ x\in B \}$.

• $S^c:=\{ x\in\Omega:x\notin S\}$, where $\Omega$ is universal set.

• $A\setminus B:=$ $\{ x\in A:x\notin B\}=A\cap B^c$

• $\mathbb{N}:= \{0, 1, 2, 3, \ldots\}=$ Set of natural numbers.

• $\mathbb{N}/\equiv(mod\ n)$ $\space:= \{ 0, 1, 2, 3, \ldots, (n-1) \}$.

• $[n]:=\{ 1, 2, 3, \ldots, n \}$, i.e., set of first $n$ counting numbers.

• $\mathbb{Z}:=\{\ldots, -2, -1, 0, 1, 2, \ldots\}=$ Set of integers.

• $\mathbb{Q}:=\{\frac{m}{n}:m,n\in\mathbb{Z}\ and\ n\neq0\}=$ Set of rational numbers.

• $\mathbb{R}:=$ Set of real numbers.

• $\mathbb{C}:=$ Set of complex numbers.

• $S^*:=S\setminus\{0\}$, where $0\in S$. For example, $\mathbb{N}^*=\{1,2,3,\ldots\}$.

• $Family\ \{x_i\}_{i\in I}\ :=$ is a map $f:I\to S$, defined as $x_i=f(i)\ \forall i\in I$ (assuming $x_i\in S\ \forall i\in I$). Note that, elements of a family $\{x_i\}_{i\in I}$ can repeat, so it is different from the set $\{x_i\in S: i\in I \}$, where all elements are distinct.
  If the index set $I$ has some inherent order, then we may view the family $\{x_i\}_{i\in I}$ as being ordered in some way, like a list or sequence. For example, if $I=[m]$, then, we may write the family as $\{x_i\}_{i=1}^m$, which is a list of $m$ elements from set $S$.

• $\bigcup_{i\in I}S_i:=\{x: x\in S_i$ for some $i\in I\}=$ set of all $x$ such that $x$ is member of at least one set $S_i\in$ family $\{S_i\}_{i\in I}$.

• $\bigcap_{i\in I}S_i:=\{x: x\in S_i$ for all $i\in I\}=$ set of all $x$ such that $x$ is common member of all sets $S_i\in$ family $\{S_i\}_{i\in I}$.

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