For each digit, this is addition modulo 2. It;s a well-known fact that addition modulo n is commutative and associative.
There's a math problem on page-34 of Pinter's "A Book of Abstract Algebra" book which is(Problem F):
If a word is is sent, but a word is received(where the
and are or ), then the error pattern is the word where:
With this motivation, we define an operation of adding words, as follows: If and are both of length
, we add them according to the rules
, , and
If and are both of length , we add them by adding corresponding digits. That is(let us
introduce commas for convenience),
Thus the sum of and is the error pattern .
The symbol will designate the set of all the binary words of length .
We will prove that the operation of word addition has the following properties on :
- It is commutative.
- It is associative.
- There is an identity element for word addition.
- Every word has an inverse under word addition.
1) Show that
3) Show that
How can I prove that the commutativity and associativity holds in this case?