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
- $=$
- $x \ne y$ means $x$ is not the same object as $y$, and is read $x$ does not equal $y$, or $x$ 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.
Identity
- $\equiv$
The symbol $\equiv$ can be used to distinguish an identity from a conditional equation, but frequently (and usually on $\mathsf{Pr} \infty \mathsf{fWiki}$) the equals sign $=$ is used instead.
The $\LaTeX$ code for \(\equiv\) is \equiv .
The $\LaTeX$ code for \(\not \equiv\) is \not \equiv .
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
- $x^\prime$
The prime 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 prime symbol can be repeated: $x' '$ and $x' ' '$, and so on, to indicate a further variation of the type of $x$, according to the appropriate definition.
The $\LaTeX$ code for \(x'\) is x' or x^\prime.
The $\LaTeX$ code for \(x' '\) is x' ' or x^{\prime \prime}.
The $\LaTeX$ code for \(x' ' '\) is x' ' ' or x^{\prime \prime \prime}.
Quotation Marks
- $``x"$
Enclosing a symbol or a collation $x$ in quotation marks provides an indication that $x$ is to be treated as a literal string.
The $\LaTeX$ code for \(``x"\) is ``x" .
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:
| \(\ds \forall n \in \Z: \, \) | \(\ds n\) | \(<\) | \(\ds \infty\) | |||||||||||
| \(\ds \forall n \in \Z: \, \) | \(\ds n + \infty\) | \(=\) | \(\ds \infty\) | |||||||||||
| \(\ds \forall n \in \Z: \, \) | \(\ds n \times \infty\) | \(=\) | \(\ds \infty\) | |||||||||||
| \(\ds \infty^2\) | \(=\) | \(\ds \infty\) |
Similarly, the quantity written as $-\infty$ is defined as having the following properties:
| \(\ds \forall n \in \Z: \, \) | \(\ds -\infty\) | \(<\) | \(\ds n\) | |||||||||||
| \(\ds \forall n \in \Z: \, \) | \(\ds -\infty + n\) | \(=\) | \(\ds -\infty\) | |||||||||||
| \(\ds \forall n \in \Z: \, \) | \(\ds -\infty \times n\) | \(=\) | \(\ds -\infty\) | |||||||||||
| \(\ds \paren {-\infty}^2\) | \(=\) | \(\ds -\infty\) |
The latter result seems wrong when you think of the rule that a negative number squared equals a positive number, but remember that infinity is not exactly a number as such.
The $\LaTeX$ code for \(\infty\) is \infty .
End of Proof (Tombstone, or Halmos)
- $\blacksquare$
The Halmos symbol is the character: $\blacksquare$ used to indicate the end of a proof.
It replaces the old-fashioned and embarrassingly uncool Q.E.D. which muggles sometimes use when pretending to be clever.
The $\LaTeX$ code for \(\blacksquare\) is \blacksquare .
End of Subproof
- $\Box$
The empty black square: $\Box$ is a variant of the Halmos symbol, and on $\mathsf{Pr} \infty \mathsf{fWiki}$ is used to denote the end of a subproof or self-contained proof section in a big proof.
It is believed that this sort of marked breakdown helps with the mental parsing of a presentation of a proof.
The $\LaTeX$ code for \(\Box\) is \Box .