Editing Concatenation (section)

==Concatenation of sets of strings==
In [[formal language|formal language theory]] and [[pattern matching]] (including [[regular expressions]]), the concatenation operation on strings is generalised to an operation on sets of strings as follows:

For two sets of strings ''S''<sub>1</sub> and ''S''<sub>2</sub>, the ''concatenation'' ''S''<sub>1</sub>''S''<sub>2</sub> consists of all strings of the form ''vw'' where ''v'' is a string from ''S''<sub>1</sub> and ''w'' is a string from ''S''<sub>2</sub>, or formally {{math|1=''S''<sub>1</sub>''S''<sub>2</sub> = {{mset| ''vw'' : ''v'' ∈ ''S''<sub>1</sub>,  ''w'' ∈ ''S''<sub>2</sub> }}}}. Many authors also use concatenation of a string set and a single string, and vice versa, which are defined similarly by {{math|1=''S''<sub>1</sub>''w'' = {{mset| ''vw'' : ''v'' ∈ ''S''<sub>1</sub> }}}} and {{math|1=''vS''<sub>2</sub> = {{mset| ''vw'' : ''w'' ∈ ''S''<sub>2</sub> }}}}. In these definitions, the string ''vw'' is the ordinary concatenation of strings ''v'' and ''w'' as defined in the introductory section.

For example, if {{math|1=''F'' = {{mset| a, b, c, d, e, f, g, h }}}}, and {{math|1=''R'' = {{mset| 1, 2, 3, 4, 5, 6, 7, 8 }}}}, then ''FR'' denotes the set of all [[chess board]] coordinates in [[algebraic notation (chess)|algebraic notation]], while e''R'' denotes the set of all coordinates of the kings' [[file (chess)|file]].

In this context, sets of strings are often referred to as formal languages.  The concatenation operator is usually expressed as simple juxtaposition (as with [[multiplication]]).