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

K^N

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.