Answer
See explanation
Work Step by Step
Below is a standard recursive definition for the set of all strings over \(\{a,b\}\) that contain **exactly one** \(a\). Let us call this set \(S\).
---
## **Recursive Definition**
1. **Base Case**
\[
a \in S.
\]
(The one‐letter string \(a\) clearly has exactly one \(a\).)
2. **Recursive Step**
If \(x \in S\), then
\[
b x \in S
\quad\text{and}\quad
x b \in S.
\]
Appending a \(b\) on either side of a string with exactly one \(a\) does not change the number of \(a\) s.
3. **Restriction**
No other strings are in \(S\) except those obtained from the base case by finitely many applications of the recursive step.
In words:
- We start with the string \(a\).
- We can insert the letter \(b\) on the left or on the right as many times as we like.
- We do **not** allow adding any additional \(a\) s; that ensures we never exceed one \(a\).
Hence, \(S\) ends up being exactly all strings of the form \(b^m a\,b^n\), where \(m,n\) are nonnegative integers—i.e., precisely those strings containing exactly one \(a\).
