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∑ is  a string alphabet  that consists of a finite set of symbols. 

Example ∑ = {a,b} or ∑ = {0,1} or ∑ = {a,b,c}

Power of Alphabet (∑):

Here $\underset{ }{\overset{ }{{\sum }^n}}$ (for some integer n) denotes the set of strings of length n with symbols from ∑. 

In other words,

$\underset{ }{\overset{ }{{\sum }^n}}$= {w | w is a string over and | w | = n}. 

Hence, for any alphabet, $\underset{ }{\overset{ }{{\sum }^0}}$ denotes the set of all strings of length zero. That is, = {e}. 

Let ∑ = {a,b} then

$\underset{ }{\overset{ }{{\sum }^0}}$ = {λ}

$\underset{ }{\overset{ }{{\sum }^1}}$ = {a, b}

$\underset{ }{\overset{ }{{\sum }^2}}$ = (a,b).(a,b) = aa, ab, ba, bb (4 string with two length over a,b)

So,  $\underset{ }{\overset{ }{{\mathbf{\sum }}^{\mathbf{n}}}}\mathbf{ }\mathbf{=}{\mathbf{ }\mathbf{2}}^{\mathbf{n}}$ number of string = | $\underset{ }{\overset{ }{{\mathbf{\sum }}^{\mathbf{n}}}}$|

• At most n length: -

$\underset{ }{\overset{ }{{\sum }^{*}}}$:- The set of all strings over an alphabet is denoted by $\underset{ }{\overset{ }{{\sum }^{*}}}$. It contains set of all the strings that can be generated by iteratively concatenating symbols from any number of times. That is,

Example: 

Let ∑ = {a,b} then

$\underset{ }{\overset{ }{{\sum }^{*}}}$is a set of all strings generated by a, b. Basically $\underset{ }{\overset{ }{{\sum }^{*}}}$is a universal set over a,b i.e. 

 (a + b)*.

Here $\underset{ }{\overset{ }{{\sum }^{*}}}$ = { ε, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, …} 

[We can write ε or λ any one]

$\underset{ }{\overset{ }{{\sum }^{+}}}$ :- The set of all strings over an alphabet excluding λ  is denoted by $\underset{ }{\overset{ }{{\sum }^{+}}}$. That is,

Example: 

Let ∑ = {a,b} then

$\underset{ }{\overset{ }{{\sum }^{+}}}$is a set of all strings generated by a and b with at least one length of the string. 

$\underset{ }{\overset{ }{{\sum }^{+}}}$ is also universal set over a,b but excluding  i.e. ${\mathbf{(}\mathbf{a}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{b}\mathbf{)}}^{\mathbf{+}}$

Here $\underset{ }{\overset{ }{{\sum }^{+}}}$ = {a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, …} 

Note :  To understand the concept of $\underset{ }{\overset{ }{{\sum }^{*}}}$ and $\underset{ }{\overset{ }{{\sum }^{+}}}$  is very important for Automata.