Answer

Correct Option: (iii) All finite strings over $\Sigma$ including $\epsilon$

Explanation

Definition: $\Sigma^* = \bigcup_{i=0}^{\infty} \Sigma^i$, where $\Sigma^0 = \{\epsilon\}$, $\Sigma^1 = \Sigma$, $\Sigma^2 = \Sigma\Sigma$, ...

For $\Sigma = \{a,b\}$:

• $\epsilon$ (empty string, length 0) ∈ $\Sigma^*$
• $a, b$ ∈ $\Sigma^*$
• $aa, ab, ba, bb$ ∈ $\Sigma^*$
• … all finite concatenations.

Properties:

• Countably infinite set.
• Does not contain infinite strings (each string is finite; there are infinitely many strings, but each has finite length).
• Not limited to equal length.

Options:

• (i) Only finite strings but missing $\epsilon$ → incomplete
• (ii) Only infinite strings → false
• (iv) Only equal length → false

Final Answer

All finite strings including $\epsilon$ → Option (iii).