Poor Example 8.1.
\begin{equation*}
3+5=8/2=4\text{.}
\end{equation*}
Section 8 Skill: Mathematical Symbols
You will encounter many mathematical symbols during your math courses. The table below provides you with a list of the more common symbols, how to read them, and notes on their meaning and usage.
| Symbol | How to read it | Notes on meaning and usage |
|---|---|---|
| \(a=b\) | \(a\) equals \(b\) | \(a\) and \(b\) have exactly the same value. |
| \(a \approx b\) or | \(a\) is approximately equal to \(b\) | Do not write \(=\) when you mean \(\approx\text{.}\) |
| \(P\Rightarrow Q\) | \(P\) implies \(Q\) | If \(P\) is true, then \(Q\) is also true. |
| \(P\Leftarrow Q\) | \(P\) is implied by \(Q\) | If \(Q\) is true, then \(P\) is also true. |
| \(P\Leftrightarrow Q\) or \(P\) iff \(Q\) | \(P\) is equivalent to \(Q\) or \(P\) if and only if \(Q\) | \(P\) and \(Q\) imply each other. |
| \((a,b)\) | the point \(a\) \(b\) | A coordinate in \(\mathbb{R}^{2}\text{.}\) |
| \((a,b)\) | the open interval from \(a\) to \(b\) | The values between \(a\) and \(b\text{,}\) but not including the endpoints. |
| \([a,b]\) | the closed interval from \(a\) to \(b\) | The values between \(a\) and \(b\text{,}\) including the endpoints. |
| \((a,b]\) | The (half-open) interval from \(a\) to \(b\) excluding \(a\text{,}\) and including \(b\text{.}\) | The values between \(a\) and \(b\text{,}\) excluding \(a\text{,}\) and including \(b\text{.}\) Similar for \([a,b)\text{.}\) |
| \(\mathbb{R}\) or \(\mathbf{R}\) | the real numbers | It can also be used for the plane as \(\mathbb{R}^{2}\text{,}\) and in higher dimensions. |
| \(\mathbb{C}\) or \(\mathbf{C}\) | the complex numbers | \(\{a+bi: a,b\in \mathbb{R}\}\text{,}\) where \(i^{2}=-1\text{.}\) |
| \(\mathbb{Z}\) or \(\mathbf{Z}\) | the integers | …, \(-2\text{,}\) \(-1\text{,}\)0,1,2,3, …. |
| \(\mathbb{N}\) or \(\mathbf{N}\) | the natural numbers | \(1,2,3,4, \ldots\text{.}\) |
| \(a\in B\) | \(a\) is an element of \(B\) | The variable \(a\) lies in the set (of values) \(B\text{.}\) |
| \(a\notin B\) | \(a\) is not an element of \(B\) | |
| \(A \cup B\) | \(A\) union \(B\) | The set of all points that fall in \(A\) or \(B\text{.}\) |
| \(A \cap B\) | \(A\) intersection \(B\) | The set of all points that fall in both \(A\) and \(B\text{.}\) |
| \(A \subset B\) | \(A\) is a subset of \(B\) or \(A\) is contained in \(B\) | Any element of \(A\) is also an element of \(B\text{.}\) |
| \(\forall x\) | for all \(x\) | Something is true for all (any) value of \(x\) (usually with a side condition like \(\forall x>0\) ). |
| \(\exists\) | there exists | Used in proofs and definitions as a shorthand. |
| \(\exists!\) | there exists a unique | Used in proofs and definitions as a shorthand. |
| \(f\circ g\) | \(f\) composed with \(g\) or \(f\) of \(g\) | Denotes \(f(g(\cdot))\text{.}\) |
| \(n!\) | \(n\) factorial | \(n!=n(n-1)(n-2)\cdots \times 2 \times 1\text{.}\) |
| \(\lfloor x \rfloor\) | the floor of \(x\) | The nearest integer \(\le x\text{.}\) |
| \(\lceil x \rceil\) | the ceiling of \(x\) | The nearest integer \(\ge x\text{.}\) |
| \(f={\cal O}(g)\) or \(f=O(g)\) | \(f\) is big oh of \(g\) | \(\lim_{x\rightarrow\infty}\sup_{y>x}|f(y)/g(y)|<\infty\text{.}\) Sometimes the limit is toward \(0\) or another point. |
| \(f=o(g)\) | \(f\) is little oh of \(g\) | \(\lim_{x\rightarrow\infty}\sup_{y>x}|f(y)/g(y)|=0\text{.}\) |
| \(x\rightarrow a^{+}\) | \(x\) goes to \(a\) from the right | \(x\) is approaching \(a\text{,}\) but \(x\) is always greater than \(a\text{.}\) Similar for \(x\rightarrow a^{-}\text{.}\) |
Subsection 8.1 The Trouble with \(=\)
The most commonly used, and most commonly misused, symbol is ‘\(=\)’. The ‘\(=\)’ symbol means that the things on either side are actually the same, just written a different way. The common misuse of ‘\(=\)’ is to mean ’do something’. For example, when asked to compute \((3+5)/2\text{,}\) some people will write:
This claims that \(3+5=4\text{,}\) which is false. We can fix this by carrying the ‘\(/2\)’ along, as in \((3+5)/2=8/2=4\text{.}\) We could instead use the ’ \(\Rightarrow\) ’ symbol, meaning ’implies’, and turn it into a logical statement:
Good Example 8.2.
\begin{equation*}
3+5=8\quad \Rightarrow \quad (3+5)/2=4\text{.}
\end{equation*}
Subsection 8.2 To Symbol or not to Symbol?
Poor Example 8.3.
\(\lim_{x \rightarrow x_0}f(x)=L\) means that \(\forall \epsilon >0\text{,}\) \(\exists \delta >0\) s.t. \(\forall x\text{,}\)
\begin{equation*}
0 < |x -x_{0}|< \delta\quad \Rightarrow\quad |f(x)-L|<\epsilon\text{.}
\end{equation*}
Although this statement is correct mathematically, it is difficult to read (unless you are well-versed in math-speak). This example shows that although you can write math in all symbols as a shortcut, often it is clearer to use words. A compromise is often preferred.
Good Example 8.4. The Formal Definition of a Limit.
Let \(f(x)\) be defined on an open interval about \(x_{0}\text{,}\) except possibly at \(x_{0}\) itself. We say that \(f(x)\) approaches the limit \(L\) as \(x\) approaches \(x_{0}\text{,}\) and we write
\begin{equation*}
\lim_{x \rightarrow x_0}f(x)=L
\end{equation*}
if for every number \(\epsilon>0\text{,}\) there exists a corresponding number \(\delta>0\) such that for all \(x\) we have
\begin{equation*}
0 < |x -x_{0}|< \delta \quad\Rightarrow\quad |f(x)-L|<\epsilon\text{.}
\end{equation*}
Subsection 8.3 Other Examples
The ‘\(\Rightarrow\)’ symbol should be used even when doing simple algebra.
Good Example 8.5.
\begin{equation*}
(y-0)=2(x-1) \quad \Rightarrow \quad y=2x-2
\end{equation*}
You will be more comfortable with symbols, and better able to use them, if you connect them with their spoken form and their meaning.
Good Example 8.6.
The mathematical notation \((f \circ g)(x)\) is read “\(f\) composed with \(g\) at the point \(x\)” or “\(f\) of \(g\) of \(x\)” and means
\begin{equation*}
f(g(x))\text{.}
\end{equation*}
