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}
• Σ¹= {0,1} ( 2¹=2)
• Σ²= {00,01,10,11} (2²=4)
• Σ³= {000,001,010,011,100,101,110,111} (2³= 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⁺= Σ¹U Σ²U Σ³…………. 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*= Σ⁰U Σ¹U Σ²U Σ³…………

={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⁺= Σ¹U Σ²U Σ³…………

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

Set of all strings excluding epsilon is called kleene plus

Σ⁺= Σ*- { ε}

Or

Σ⁺= Σ¹U Σ²U Σ³