Suppose that two binary operations, denoted by and , are defined on a nonempty set S, and that the following conditions are satisfied .

(1) and are in S

(2) and

(3)

Also, and , the elements and are defined recursively as follows:

if and have been defined, then and

Which of the following must be true?

(i) and

(ii) and

(ii) and

This is one obviously true.

I am struggling with proving or disproving 1 and 2