If A and B are finite sequences and A , B = B , A...

Michelle Mendoza

Michelle Mendoza

Answered

2022-07-07

If A and B are finite sequences and A , B = B , A (where the comma denotes the concatenation of sequences), then both A and B are repetitions of some sequence C.

Answer & Explanation

Rafael Dillon

Rafael Dillon

Expert

2022-07-08Added 15 answers

Step 1
If A,B are strings of length | A | , | B | 1 and A B = B A then there is some string D such that A = D k , B = D l with k , l { 0 , 1 , . . }
Suppose that when A,B are strings of length | A | , | B | n and A B = B A then there is some string D such that A = D k , B = D l with k , l { 0 , 1 , . . }.
Step 2
Now suppose | A | n + 1 , | B | = n + 1 and A B = B A. If | A | = | B | then A = B and we can take D = A, so suppose | A | n. Then we must have B = A C for some | C | 1. Then A A C = A C A and so A C = C A and hence there is some D such that A = D k , C = D l . Then A = D k , B = D k + l .

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