Power Of Sigma (Σ) in TOC
Sigma is set of symbols, which are always finite. It is denoted with “Σ”. Example of Sigma with 4 values is given below
Σ = {a,0,b,k}
Where a, 0, b, k are the values of sigma.
Formula of Powers of ‘ Σ ‘
Power of sigma follows the formula
KN
Where K tells about the total number of Sigma values and N is any positive Integer which tells about the maximum length of string.
if K=4 and N=2 then it means there are 4 sigma values i.e. Σ(a,b,c,d) and Power 2 tells every string of language may contains maximum length of 2 as given below
Σ(a,b,c,d)4 = Σ4 = 42 = 16
It means there are 16 strings over Σ of length 2. i.e.{aa, bb, cc, dd, ab, ac, ad, ………} where ‘aa’ is a string of length 2.
Examples of Power Of Sigma
Example 1: Suppose there are 2 alphabets of sigma i.e. (Σ= {a,b}). Then power of sigma is given bellows
Sigma with power Zero.
Σ0 = Set of all (Σ0 = 20 = 1) zero strings over Σ of length 0. {ε} // no element called Absalom or lamda
Sigma with Power 1.
Σ(a,b)1 = Σ1 = 21 = 2
It means there are only two strings over Σ of single length. i.e.{a, b} where ‘a’ is a single length
Sigma with Power 2
Σ(a,b)2 = Σ2 = 22 = 4
It means there are only four strings over Σ of length 2. i.e.{aa, ab, ba, bb} where ‘aa’ is a double length
Sigma with Power 3
Σ(a,b)3 = Σ3 = 23 = 8
It means there are only 8 strings over Σ of length 3. i.e. {aaa, aab, baa………bbb} where ‘ aaa’ is a triple length.
In the same way, sigma with power 4, 5, 6 and so on.