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Mathematical Preliminaries: Mathematical Symbols

Learning Outcomes

Identify the basic symbols of mathematics that will be used in the lecture
Define sets
Define functions

Mathematical Symbols

The following are the mathematical symbols used throughout these online material.

Curly brackets are used to designate sets. For example $A=\{1,2,3\}$ is the set of the three elements 1, 2, and 3.
We use the symbol $\in$ to designate elements in sets. For example, if $A=\{1,2,3\}$ , then $1\in A$ .
The symbol $\subset$ means “subset of” and is used to indicate that the elements of one set are elements of another set. For example, if $A=\{1,2,3\}$ and $B=\{1,2,3,4,5,6\}$ , then $A\subset B$ .
Sets can be defined using properties listed within the curly brackets. For example, we can define the set $A=\{x | x\mbox{ is a day of the week}\}$ . This means that $A=\{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday\}$ .
$\mathbb{N}$ is used to designate the set of natural numbers starting from 1. i.e., $N=\{1,2,3,\cdots\}$ .
$\mathbb{Z}$ is used to designate the set of integers. i.e., $\mathbb{Z}=\{\cdots,-3,-2,-1,0,1,2,3,\cdots\}$ .
$\mathbb{Q}$ is used to designate the set of rational numbers. i.e., $\mathbb{Q}=\left\{\frac{a}{b}|a,b\in \mathbb{Z},b\neq 0\right\}$ .
$\mathbb{R}$ is used to designate the set of real numbers.
Box brackets are used to designate (closed) intervals in real numbers. For example, $A=[0,1]$ is the set of real numbers between 0 and 1 and including both 0 and 1.
Round brackets are used to designate (open) intervals in real numbers. for example, $B=(0,1]$ is the set of real numbers between 0 and 1 and including 1.
The symbol $\forall$ means “for every” and is usually used to designate a property of all the elements of a set. For example, if we say that $A=\{1,2,3\}$ , we can describe the fact that all the elements in $A$ are greater than zero by stating: $\forall x\in A:x>0$ .
The symbol $\exists$ means “exists” and is used to identify particular elements in sets. For example, if $A=\{1,2,3\}$ we can say that $\exists x\in A$ such that $x>0$ , i.e., there exists an element in the set $A$ which is greater than 0.
The symbol $\exists!$ means “exists and unique” and is used to identify a unique element in a set. For example, if $A=\{1,2,3\}$ we can say that $\exists! x\in A$ such that $x>2$ , i.e., there exists one element and one element only in the set $A$ which is greater than 2.
To define functions or maps between sets, we use $\rightarrow$ . If $A$ is a set and $B$ is another set, then a function that maps elements of $A$ into elements of $B$ is defined with the notation $f:A\rightarrow B$ .
If $f:A\rightarrow B$ , then, the arrow $\mapsto$ is used to designate that a function $f$ maps the element $x\in A$ to $f(x)\in B$ as follows $f:x\in A\mapsto f(x)\in B$ .
The symbol $\times$ is used in many contexts. One of them is the cross product which will be defined later. Another is for defining the set of ordered pairs. For example, if $A$ and $B$ are two sets. Then, the set $C=A\times B$ is the set made of ordered pairs $(x,y)$ with $x\in A$ and $y\in B$ . For example if $A=\{1,2\}$ and $B=\{3,4,5\}$ , then $C=A\times B=\{(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)\}$ . The set $\mathbb{R}^2=\mathbb{R}\times \mathbb{R}$ is the set of ordered pairs of real numbers and geometrically designates a plane. Similarly, $\mathbb{R}^3=\mathbb{R}\times \mathbb{R}\times \mathbb{R}$ is the set representing the three dimensional space.
$\mbox{Statement 1}\Leftrightarrow \mbox{Statement 2}$ is used to designate equivalence of statements 1 and 2(if and only if). For example: $A\subset \mathbb{R}$ is equivalent to $\forall x\in A:x\in \mathbb{R}$ . So, this can be written as: $A\subset \mathbb{R} \Leftrightarrow \forall x\in A:x\in \mathbb{R}$ .
$\mbox{Statement 1}\Rightarrow \mbox{Statement 2}$ is used to describe that statements 1 leads to statement 2. For example: $A=\{1,2,3\}$ leads to the statement $A\subset \mathbb{N}$ . So, this can be written as: $A=\{1,2,3\} \Rightarrow A\subset \mathbb{N}$ .
The Kronecker delta $\delta_{ij}$ is a special symbol which gives a value of either 0 or 1 depending on the subscripts $i$ and $j$ . Usually $i$ and $j$ can take values of 1, 2, or 3 and
$\delta_{ij}= \begin{cases} 1 & i=j\\ 0 & i\neq j \end{cases}$
The alternator $\varepsilon_{ijk}$ is a special symbol which gives a value of either 0, 1, or -1 depending on the subscripts $i$ , $j$ , and $k$ . Usually $i$ , $j$ , and $k$ can take values of 1, 2, or 3 and
$\varepsilon_{ijk}= \begin{cases} 1 & i,j,k \mbox{ is a cyclic permutation of 1, 2, 3}\\ -1 & i,j,k \mbox{ is a noncyclic permutation of 1, 2, 3}\\ 0 & \mbox{Otherwise} \end{cases}$
Unless stated explicitly, we do not differentiate between the column or row representation of vectors in $\mathbb{R}^n$ . The curly brackets are also used for the row representation of vectors to be consistent with how Mathematica defines a vector. Therefore, the following are equivalent for the three dimensional vector $x$ with components $x_1$ , $x_2$ , and $x_3\in\mathbb{R}$ :
$x=\left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right)=\{x_1,x_2,x_3\}=(x_1,x_2,x_3)$
The reader should differentiate between vectors and sets depending on the context.

Video

Quiz 1 – Mathematical Symbols

Page Comments

Yibo Tang says:

August 31, 2021 at 6:49 pm

very useful for the start

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Kenanao Sithole says:

November 5, 2022 at 12:01 pm

Great summary

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Kamal Mohamed Noor says:

January 27, 2023 at 3:58 pm

Great effort

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