Answer Happy

1 and 2. Discrete math/structures
Consider the alphabet Σ = {x,y,z}. Recall that a strings of length k over Σ can be written as s = S1S2...Sk where each si E Σ. Define the set S recursively as follows: (Basis Step) The strings xyz,yxzxzy are in S. (Recursive Step) For any w E S, the strings xyzw.zywz,xwyz,wxyz are all in S. (i) Find all elements in S of length less than or equal to 8. (ii) Prove by structural induction that every element w E S has a length /(w) divisible by 3. [Hint: If a number m E N is divisible by 3, then there exists an n E N such that m = 3n.]