Question

this is from my theory of computation class i need this question is detail ans,

Find a simple and nontrivial characterization of the language {111} ∗{11111} ∗ . Give a proof of the correctness of your answer.

Answer 1

We are given the language {111}*{11111}* which can also be written as {111+11111}* = $\{1^3+1^5\}^*$ .

So any string from the language can be written as $1^z$ where $z=3x+5y$ and $x$ and $y$ are non-negative integers.

The set $\{z|z=3x+5y,\ x,y\in \mathbb{Z} \and\ x,y\geq 0\}$ contains all non-negative integers except 1,2,4,6 and 7.
This is because $gcd(3,5)=1$ and so we must have integers x and y ​​​satisfying $z=3x+5y,\ \forall z \in \mathbb{Z}$ . However if we restrict the domain of x and y to non-negative integers we are not getting some integers, and to be precise the integers $\{1,2,4,6,7\}$ .

So, we can characterize the language as $1^*-\{\epsilon, 1, 11, 1111, 111111, 1111111\}$ .