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 .

For example,

and

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?