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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]