Understanding Σ* and Σ in formal languages

Question

If I have Σ={a} , what words does Σ* has ?

Σ*= {a,aa,aaa,aaaa.....} ?

Thanks

Answer 1

If your alphabet is Σ={a} then Σ*= {#, a,aa,aaa,aaaa.....} means all the possible n* a, including the empty string # (phi). Another way to produce that sequence is using grammars:

S -> S
S -> aS
S -> #

where # is the empty string.

Answer 2

It has the empty string, which you didn't mention, it also contains sequences of a, of all lengths.

You can find more information at http://en.wikipedia.org/wiki/Kleene_star.

Answer 3

The * in Σ* usually denotes zero or many times. So Σ* will have the empty string, and any combination of letters from the alphabet Σ.

(Since your alphabet only has a , then Σ* will have any combination of as and the empty string.)

If your alphabet had more values i.e. Σ = {a,b} then you would have any combination of as and bs and the empty string. i.e. Σ* = {phi, a, b, aa, ab, ba, bb, bab, ...(etc)}

Answer 4

Σ* is the set of strings of any length that you can make by concatenating any number of symbols drawn from Σ (including none).

Here is one way to define Σ*:

Let Σ^n be the set of strings of length n over Σ.

Then Σ* = Σ^0 union Σ^1 union ...

Σ^0 = {phi} since phi is the only string of length 0. Therefore phi is always in Σ* no matter what Σ is.