I defined the reverse w^R of a string w by saying just write the symbols of w in opposite order. There is a more precise way of defining the reverse that helps us prove useful results. This definition is given by the following two rules:

for any alphabet symbol x, define x^R = x

for any string w and any alphabet symbol x, define (wx)^R = xw^R

Using this new definition,

a) prove that (w₁w₂)^R = w₂^R w₁^R for any two non-empty strings w₁, w₂

b) For all k 2, prove that (w₁ w_K)^R = w_k^R w₁^R for any k non-empty strings w₁, . . . , w_K.