Notation
Greek alphabet
| Symbol |
Name |
Common use |
| \(\Sigma\) |
Sigma |
Set of alphabet symbols |
| \(\Gamma\) |
Gamma |
Set of stack/tape symbols |
| \(\alpha\) |
alpha |
|
| \(\beta\) |
beta |
|
| \(\gamma\) |
gamma |
|
| \(\delta\) |
delta |
Transition function |
| \(\varepsilon\) |
epsilon |
Empty string |
| \(\sigma\) |
sigma |
|
Strings
| Notation |
Meaning |
| \(\varepsilon\) |
Empty string (Length = 0) |
| \(w\) |
String made of symbols from \(\Sigma\) |
| \(w^R\) |
String obtained by writing \(w\) in the reverse order |
| \(\mid w\mid\) |
Length of the string \(x\) |
| \(xy\) |
String made by concatenating \(x\) and \(y\) |
| \(w^n\) |
String made by concatenating \(n\) copies of \(w\): \(\underbrace{ww\ldots w}_{n \text{ copies}}\) |
|
In particular: \(w^0=\varepsilon\), \(w^1=w\) and \(w^2=ww\) |
| \(\{0,1\}^n\) |
Binary strings of length exactly \(n\) symbols |
| \(\{0,1\}^*\) |
Binary strings of any length: \(\{\varepsilon, 0, 1, 00, 01, 10, 11, 000, \ldots\}\) |
Regular expressions
The symbol \(■\) is just a place holder.
| Notation |
Meaning |
| \(■ + ■\) |
Union ("or") |
| \(■\,■\) |
Concatenation ("gluing" two strings) (juxtaposition/no symbol) |
| \(■^*\) |
Star (zero or more copies) e.g. \(1^*=\{\varepsilon,1,11,111,1111,\ldots\}\) |
| \(■^+\) |
One or more copies -- shorthand for \(■\,■^*\) e.g. \(1^+=\{1,11,111,1111,\ldots\}\) |
| \(\Sigma^*\) |
Any string of finite length over the given alphabet, including \(\varepsilon\) (zero length) |
| \(\Sigma^+\) |
Any string of finite non-zero length over the given alphabet (not \(\varepsilon\)) |
| \(()\) |
Grouping, to override usual precedence rule: star, concatenation, union |
| \(\varepsilon\) |
Empty string |
| \(\emptyset\) |
No strings at all |
Sets and logic notation
| Notation |
Meaning |
| \(\{{\textcolor{gray}{x_1,\ldots,x_n}}\}\) |
Finite set consisting of the elements \(x_1\) until \(x_n\) |
| \(\{{\textcolor{gray}{pattern}} \mid {\textcolor{gray}{condition}}\}\) |
Set of items matching \(pattern\) and satisfying \(condition\). |
|
The "\(\mid\)" symbol is read "such that" |
| \(\emptyset\) |
Empty set, i.e. \(\{\}\) |
| \(\in\) |
"in", member of a set |
| \(\notin\) |
"not in", not a member of a set |
| \(\cup\) |
Union of two sets |
| \(\cap\) |
Intersection of two sets |
| \(-\) |
Difference of two sets |
| \(\times\) |
Cartesian product of two sets |
| \(\subset\) |
Subset of ... |
| \(\mid A\mid\) or \(\#A\) |
Cardinality of the set \(A\), i.e. count of its elements |
| \(2^A\) |
Power set of \(A\), i.e. set of all subsets of \(A\) |
| \(\land\) |
Logical "and" |
| \(\lor\) |
Logical "or" |
| \(\lnot\) or \(\bar{\textcolor{gray}{■}}\) |
Logical "no" |
| \(\oplus\) |
Logical "xor" -- "exclusive or" |
| Notation |
Meaning |
| \(\mathbb{N}\) |
Natural numbers: \(\{1,2,3,\ldots\}\) |
| \(\mathbb{Z}\) |
Integers: \(\{0,1,-1,2,-2,3,-3,\ldots\}\) |
| \(\mathbb{Z}_{\geq 0}\) |
Non-negative integers: \(\{0,1,2,3,\ldots\}\) |
| \(S'\) |
A set called "\(S\) prime" (a way of making new names) |
| \(S''\) or \(S'''\) |
A set called "\(S\) double prime" / "\(S\) triple prime" |
Numeric
| Notation |
Meaning |
| \(=\) |
equals |
| \(\neq\) |
not equal |
| \(<\) |
less than |
| \(\leq\) |
less than or equal |
| \(>\) |
greater than |
| \(\geq\) |
greater than or equal |
| \(n!\) |
Factorial of \(n\): \(n\times(n-1)\times (n-2)\times \cdots \times 2 \times 1\) |