**Power Of Sigma **(**Σ**)** in TOC**

**Sigma **is a set of symbols that are always ** finite**. It is denoted with “

**”. An example of Sigma with four values is given below.**

__Σ__Σ = {a,0,b,k} Where a, 0, b, and k are sigma values.

**Powers of ‘ Σ ‘ Formula**

The following is the formula of Power of Sigma

K^{N}

Where **K** tells about the total number of **Sigma values**, and **N** is any positive Integer that tells about the **maximum length of the string**.

**Power Of Sigma ****Examples **

Following are the various examples of the power of Sigma. Suppose there are** two alphabets of Sigma**, i.e. (__Σ= {a,b}__). Hence, **K=2. **The power of Sigma with different values is given below.

**Sigma with power Zero (K=2, N=0)**

**Σ**** ^{0}** =

**Σ**

^{0 }**=**

**2**

^{0}**=**1

Set of all strings over Σ of length 0 is only epsilon {ε}.

**Sigma with Power 1 ****(K=2, N=1)**

**Σ(a,b)**** ^{1}** =

**Σ**

^{1 }**=**

**2**

^{1}**=**2

It means there are only two strings over Σ of a single length. i.e.{a, b} where ‘a’ is a single length

**Sigma with Power 2 ****(K=2, N=2)**

**Σ(a,b)**** ^{2}** =

**Σ**

^{2 }**=**

**2**

^{2}**=**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 ****(K=2, N=3)**

**Σ(a,b)**** ^{3}** =

**Σ**

^{3 }**=**

**2**

^{3}**=**8

It means there are only eight 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.**

**Power of Sigma when ****(K=4, N=2)**

**If K=4 and N=2,** then it means there are four sigma values, i.e., Σ(a,b,c,d), and Power 2 tells every string of language may contain a maximum length of 2 as given below

**Σ(a,b,c,d)**** ^{2}** =

**Σ**

^{2 }**=**

**4**

^{2}**=**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.

Note:Similarly, Unlimited cases can be seen by changing the values of sigma and its power

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