String Concatenation (Operation)

• Definition: Concatenation is a binary operation on strings where two strings are joined end-to-end.
• Notation:
  • Concatenation ($\cdot$): For strings $x,y\in {\sum }^{\ast }$, the concatenation is written as $xy$, meaning the string formed by appending $y$ to $x$.
    • Example: $a\cdot b=ab,ab\cdot ba=abba$
  • Exponentiation of a string: For $w\in {\sum }^{\ast }$ and $n\in \mathbb{N}$:

$$w^n=\{\begin{array}{ll}\epsilon & n=0\\ w\cdot w^{n-1}& n>0\end{array}$$

Example: $(ab{)}^2=ab\cdot ab=abab$

• Properties:
  • Associative: $(xy)z=x(yz)$.
  • Identity Element: The empty string $\epsilon$, since $x\epsilon =\epsilon x=x$.
  • Not commutative: In general, $xy\ne yx$
• Example:
  • “ab” concatenated with “cd” gives “abcd”.