solution A to 1.2(3) by Dr. Gene Chase. January 31, 2008. Version 1.0 To prove: for all strings w , G*, (wR)R = w. Let's prove it by induction on the length n of the string w. (1)
First, I see that the definition of reverse is given informally on p. 17. Here's a formal definition, which--as we should expect--requires recursion to prevent our needing to use "..." Definition: The reverse of b, written bR is defined as follows: 1. If b = 8 then define bR = 8. 2. If b 8 then b can be written as xa where a is of length 1 and x may or may not be 8. In that case, define bR as axR. That is, as a concatenated with the reverse of x. OK, now we can proceed with the proof, the two parts of which will use closely the two parts of the definition of reverse. 1. Base case: n=0. If w has length 0, then w = 8. By definition of reverse, 8R = 8, (2) (8R)R = 8R by substituting from equation (2) in the left hand side to get the r.h.s. (3) R R So substituting from again (2), (8 ) =8, which is what we were to prove when w = 8. 2. Inductive assumption: Assume for all k # n that if w is of length k, then (wR)R = w. Now consider a string w of length n+1. If it's of length n+1, then it's of the form ua where u is of length n and a is of length 1: w = ua. (Compare carefully what your author did on line 7 of page 18!) (4) So the right hand side of equation (1) transforms as follows: (wR)R = ((ua)R)R = ((auR)R = ((uR)Ra) = ua =w Q.E.D. Notice that we were able to prove it without knowing what the specific alphabet G was. That's not to say that it's not important, only that I proved the theorem for all alphabets G. by substitutiion from equation (4) by the second clause of the definition of reverse again by the second clause of the definition of reverse, but this time the string uR is playing the role of x in the definition. Because the length of u is n, (uR)R = u by inductive assumption. by equation (4) again.