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# Explain the power of an alphabet in TOC.

If Σ is an alphabet, the set of all strings can be expressed as a certain length from that alphabet by using exponential notation. The power of an alphabet is denoted by Σk and is the set of strings of length k.

For example,

- Σ ={0,1}
- Σ
^{1}= {0,1} ( 2^{1}=2) - Σ
^{2}= {00,01,10,11} (2^{2}=4) - Σ
^{3}= {000,001,010,011,100,101,110,111} (2^{3}= 8)

The set of strings over an alphabet Σ is usually denoted by Σ*(Kleene closure)

For instance, Σ*= {0,1}*

={ ε,0,1,00,01,10,11,………}

Therefore, Σ*= Σ0U Σ1U Σ2U Σ3…………. With ε symbol

The set of strings over an alphabet Σ excluding ε is usually denoted by Σ^{+}(Kleene plus)
For instance, Σ^{+}={0,1}^{+}

={0,1,00,10,01,11,…………}

Therefore, Σ^{+}= Σ*- { ε}

Or

Σ^{+}= Σ^{1}U Σ^{2}U Σ^{3}…………. Without ε symbol

The power of alphabet is of two types, which are explained below −

- Kleene closure (Σ*)
- Kleene plus (Σ
^{+})

## Kleene Closure: Σ*

Let Σ ={a,b}

Σ*= Σ^{0}U Σ^{1}U Σ^{2}U Σ^{3}…………

={ε} U {a,b} U {aa,ab,ba,bb}...........

Set of all strings including epsilon is called Kleene closure

## Kleene Plus:Σ^{+}

Let Σ ={a,b}

Σ^{+}= Σ^{1}U Σ^{2}U Σ^{3}…………

={a,b} U {aa,ab,ba,bb}...........

Set of all strings excluding epsilon is called kleene plus

Σ^{+}= Σ*- { ε}

Or

Σ^{+}= Σ^{1}U Σ^{2}U Σ^{3}

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