# Symbols:General

## Symbols for General Use

### Ellipsis

- $\ldots$ or $\cdots$

An **ellipsis** is used to indicate that there are omitted elements in a set or a sequence whose presence need to be inferred by the reader.

For example:

- $1, 2, \ldots, 10$

is to be understood as meaning:

- $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$

There are two forms of the horizontal **ellipsis**, one on the writing line which is to be used for punctuation separated lists:

- $a, b, \ldots, z$

and one centrally placed in the line, to be used in other circumstances, for example, in expressions assembled using arithmetic operations:

- $a + b + \cdots + k$

There also exist vertically and diagonally arranged ellipses, for use in the structure of matrices:

- $\begin{array}{c} a \\ \vdots \\ b \end{array} \qquad \begin{array}{c} a \\ & \ddots \\ & & b \end{array}$

The $\LaTeX$ code for \(1, 2, \ldots, 10\) is `1, 2, \ldots, 10`

.

The $\LaTeX$ code for \(1 + 2 + \cdots + 10\) is `1 + 2 + \cdots + 10`

.

The $\LaTeX$ code for \(\vdots\) is `\vdots`

.

The $\LaTeX$ code for \(\ddots\) is `\ddots`

.

### Equals

- $=$

- $x = y$ means
**$x$ is the same object as $y$**, and is read**$x$ equals $y$**, or**$x$ is equal to $y$**.

- $x \ne y$ means
**$x$ is not the same object as $y$**, and is read**$x$ is not equal to $y$**.

The expression:

- $a = b$

means:

- $a$ and $b$ are names for the same object.

The $\LaTeX$ code for \(=\) is `=`

.

The $\LaTeX$ code for \(\ne\) is `\ne`

or `\neq`

.

### Negation

- $\not =, \ \not>, \ \not<, \ \not \ge, \ \not \le, \ \not \in, \ \not \exists, \ \not \subseteq, \ \not \subset, \ \not \supseteq, \ \not \supset$

The above symbols all mean the opposite of the non struck through version of the symbol.

For example, $x \not\in S$ means that $x$ is not an element of $S$.

The slash $/$ through a symbol can be used to reverse the meaning of essentially any mathematical symbol (especially relations), although it is used most frequently with those listed above.

The $\LaTeX$ code for negation is `\not`

followed by the code for whatever symbol you want to negate.

For example, `\not \in`

will render $\not \in$.

Note that several of the above relations also have their own $\LaTeX$ commands for their negations, for example `\ne`

or `\neq`

for `\not =`

, and `\notin`

for `\not \in`

.

### Prime

- $x'$

The symbol $'$ is a general indicator of **another version of** or **another type of** where the specific version or type that is being described is to be defined.

The symbol $x'$ should technically be voiced **x prime**, although colloquially referred to as some variant of **x dash** or **x tick** or whatever can be devised by the ingenuity of the reader.

The $\LaTeX$ code for \(x'\) is `x'`

or `x^{\prime}`

.

### Infinity

- $\infty$

Informally, the term **infinity** is used to mean **some infinite number**, but this concept falls very far short of a usable definition.

The symbol $\infty$ (supposedly invented by John Wallis) is often used in this context to mean **an infinite number**.

However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about.

It is defined as having the following properties:

- $\forall n \in \Z: n < \infty$

- $\forall n \in \Z: n + \infty = \infty$

- $\forall n \in \Z: n \times \infty = \infty$

- $\infty^2 = \infty$

Similarly, the quantity written as $-\infty$ is defined as having the following properties:

- $\forall n \in \Z: -\infty< n$

- $\forall n \in \Z: -\infty + n = -\infty$

- $\forall n \in \Z: -\infty \times n = -\infty$

- $\paren {-\infty}^2 = -\infty$

The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such.

The $\LaTeX$ code for \(\infty\) is `\infty`

.