Use mathematical induction to prove the following statement:

Consider an alphabet S = {a,b}. Recall that S^n is the set of all strings over S of length n. Define S^{<=n} as union of S^i for i<=n. Then S^{<=n} has (2^{n+1} - 1) elements.

1. Identify P(n), the property on natural numbers to prove the statement.

2.0. Write the base case of induction.

2.1. Prove the base case of induction.

3. Write the induction hypothesis.

4. Prove the induction step.