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Definition: Strings (words) over an Alphabet

From a formal point of view, an alphabet is a non-empty set, denoted by the Latin capital letter \(\Sigma \) (“sigma”), whose elements are called letters.

We define the concatenation of any to letters $x,y\in\Sigma$ and denote this operation by the multiplication sign “\(\cdot\)”.

Using the concatenation, we can create words, also called strings \(s\) over the alphabet \(\Sigma\).

We denote by \(\Sigma^*\) the set of all strings over the alphabet \(\Sigma \).

Let $s\in\Sigma^*$ be a string. The length of the string, denoted by \(|s|\), is defined as the number of concatenated letters which were used to create the string $s$.

Please note that $|s|\ge 0$ for any $s\in\Sigma^*$. If \(|s|=0\), then we call $s$ the empty string and denote it by \(\epsilon\).

| | | | | created: 2014-06-22 15:42:14 | modified: 2020-05-09 21:01:11 | by: bookofproofs | references: [577]

1.Example: Examples of Strings over Alphabets

2.Corollary: Algebraic Structure of Strings over an Alphabet

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[577] Knauer Ulrich: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001