Bit Strings: Recursive Definition

[sta|ker]

Homework Statement

Give a recursive definition for the set
of bit strings \{ 0^{r} 1^{s} 0^{r} \| r, s \in N \}. Note the number of 0’s must be equal, but the
number of 1’s may be different from the number of 0’s.

Homework Equations

N/A

The Attempt at a Solution

I believe this is:
Basis: \lambda \in A
Recursive Step: If \omega \in A, then 0 \omega 1 \omega 0 \in A

I just want to verify if this is correct or not.

Thanks!

 

[Office_Shredder]

If \omega = 010 then you just claimed that 001010100 is in A.

 

[sta|ker]

If \omega = 010 then you just claimed that 001010100 is in A.

Ohh, I think I see. w represents the initial value, which would basically make everything I put wrong. What about the following:

Basis: 0 1^{n} 0 \in A , n \in N
Recursive Step: if w \in A, then 0 w 0 \in A

Because, this would give: A = \{ 010, 00100, 001100, ... \}