The reversal of a string w, denoted w^R, is the string "spelled backwards". For example (reverse)^R = esrever. A careful definition can be given by induction on the length of
a string:
1. If w is a string of length 0, then w^R = epsilon
2. If w = ua, where a is an element of all alphabet and u is an element of (alphabet)*, is a string of length n+1>0, then w^R = au^R/
Prove the following by induction
(w^R)^R = w
To prove that (w^R)^R = w using induction, we will first need to establish the base case and then the inductive step.
Base Case:
For a string w of length 0, the reversal will be an empty string denoted as epsilon. Therefore, (epsilon^R)^R is equal to epsilon, which satisfies the equation (w^R)^R = w for the base case.
Inductive Step:
Assuming the equation (w^R)^R = w holds for a string of length n, we need to prove that it also holds for a string of length n + 1.
Let w = ua, where a is the last character in the string w, and u is the substring consisting of the first n characters. By the definition given, w^R = au^R.
Now we want to find (w^R)^R, which is essentially reversing the reversed string w^R. Substituting the definition of w^R:
(w^R)^R = (au^R)^R
Since the reversal operator ^R is defined recursively, we can apply the definition to each part of the expression:
(w^R)^R = (au^R)^R
= (u^R)(a^R)
Recall that a^R represents the reversal of a single character, and reversing it again will give us back the original character:
(u^R)(a^R) = (u^R)(a)
= ua
Therefore, we have shown that (w^R)^R = w holds for a string of length n + 1.
By the principle of mathematical induction, we have established that (w^R)^R = w for all strings w.
To prove the equation (w^R)^R = w by induction, we will need to show that it holds for the base case and then assume it holds for a string of length n, and show that it also holds for a string of length n+1.
Base Case:
If w is a string of length 0, then w^R = ε (the empty string). Thus, (w^R)^R = ε^R. By definition, ε^R = ε. Therefore, the equation (w^R)^R = w holds for the base case.
Inductive Hypothesis:
Assume that for a string of length n, (w^R)^R = w holds true.
Inductive Step:
Consider a string w = ua of length n+1, where a is an element of the alphabet and u is an element of (alphabet)*. We want to prove that (w^R)^R = w.
Using the definition of w^R, we have w^R = a u^R. Taking the reverse of w^R, we obtain (w^R)^R = (a u^R)^R.
By the inductive hypothesis, we know that u^R = u. Substituting this into the equation, we have (w^R)^R = (a u)^R.
Using the property that (xy)^R = y^R x^R, we can rewrite the equation as (w^R)^R = u^R a^R.
Again, using the property that x^R = x for any string x, we get (w^R)^R = u a^R.
Finally, using the definition of w^R, we can rewrite (w^R)^R as (u a^R)^R = (ua)^R.
By the definition of w^R, we know that (ua)^R = w. Therefore, we have shown that (w^R)^R = w for a string of length n+1.
By the principle of mathematical induction, we have proven that (w^R)^R = w for all strings w.