Formal definitions of alphabets and strings:
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• A symbol is an undefined term. Intuitively, a symbol is an atomic icon.
• An alphabet is a finite, nonempty set of symbols.
• An ordered n-tuples, for some n  0, over an alphabet  is called a string of length n over the
alphabet .
• In other words, any string of length n over an alphabet  is a member of the n-fold Cartesian
[n]
[3]
product of  or  . (For example, we denote  by  )
• For example, Let  = {a, b}. Then the triple (b, a, a) is a string of length 3 over  since
(b, a, a)  .
• The 0-tuple () is called the empty string.
[n]
• It follows that  for some n  0 is the set of all strings of length n over the alphabet . Note
[1]
[0]
that  = {():   } is the set of all string of length 1 over , and  = {()} has only one
member: the empty string.
• Since a symbol is atomic, we can in most cases write a string by omitting commas and ()
without confusion. For example, we write the string baa instead of (b, a, a). We use a special
[1]
symbol  for the empty string. It also follows that  becomes indistinguishable from  so
they are considered identical by natural isomorphism. Thus,  is both an alphabet and the set
of all strings of length 1 over itself.