Let X = {a, b}. A palindrome over X is a string tv for which α = αR (i.e., a string that reads the same forward and backward). An example of a palindrome over X is bbaabb. Define a function from X* to the set of palindromes over X as f (α) = ααR. Is f one-to-one? Is f onto? Prove your answers.
Let L be the set of all strings, including the null string, that can be constructed by repeated application of the following rules:
If α ϵ L. then aα b ϵ L and bαa ϵ L.
If α ϵ L and ß ϵ L, then αß ϵ L.
For example, ab is in L, for if we take α = λ, then α ϵ L and the first rule slates that ab = aαb ϵ L. Similarly, ba ϵ L. As another example, aabb is in L, for if we take α = ab, then α ϵ L; by the first rule, aabb = aαb ϵ L. As a final example, actbbba is in L, for if we take α = aabb and ß = ba, then α ϵ L and ß € L; by the second rule, aabbba = αß ϵ L.
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