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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}
- 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