Operations of String are as follows

• Concatenation
• Substring
• Reversal

• The concatenation of strings x and y, which are denoted by x×y or x y, is a string z such that:

-    z(i) = x(i) for 1 ≤ i ≤ length(x)

-    z(i) = y(i) for length(x)<i ≤length(x)+length(y)

• Example

-    automata×computation = automatacomputation

        Concatenation of string x for n times, where n≥0, can be denoted by xn

-    x0 = e

-    x1 = x

-    x2 = x x

-    x3 = x x x

-     ...

Substring

Assume x and y be strings over the alphabet Σ

The string x is a substring of y if there is strings w and z over Σ so that y = w x z.

-     ε is a substring of every string.

-     For the each string x, x is a substring of x itself.

Example

-    ε, comput and computation are the substrings of computation.

Reversal

Assume x be a string over an alphabet Σ

The reversal of string x, can be denoted by x r, is a string such that

-     if x is ε, then xr -- ε.

-     If a is in Σ, y is in Σ* and x = a y, then xr = yr a.

(automata)r

= (utomata)r a

= (tomata)r ua

= (omata)r tua

= (mata)r otua

= (ata)r motua

= (ta)r amotua

= (a)r tamotua

= (ε)r atamotua

• = atamotua The set of strings created from any number (0 or 1 or ...) of symbols in an alphabet Σ is denoted by Σ*.

• That is, Σ* = ∪i=∞0 Σi

-     Let Σ= {0, 1}.

-     Σ* = {ε, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, ... }.

• The set of strings created from at least one symbol (1 or 2 or ...) in an alphabet Σ is denoted by Σ+.

• That is, Σ+ = ∪i=∞1 Σi

=  ∪i..∞Σi-Σ0 

=  ∪i..∞Σi-{ε}

• Let  Σ = {0, 1}. S+ = {0, 1, 00, 01, 10, 11, 000, 001, 010, 011, ...}.

Σ* and Σ+ are infinite sets.

• A language over an alphabet Σ is a set of strings over Σ.

-     Let Σ = {0, 1} -- alphabet.

-     Le = {εωΣ* | number of 1's in w is even}.

-     e, 0, 00, 11, 000, 110, 101, 011, 0000, 1100, 1010, 1001, 0110, 0101, 0011, ... are in Le

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