i start by showing that there is a one to one correspondence between
and
. i take the function
where
. By the Chinese Remainder Theorem it seems very clear to me that this function is bijective. Also, this function is well defined since
along with the uniqueness of the Chinese Remainder Theorem easily implies that
.
however my TA has told me that my function is not injective and encouraged me to instead define my function
where
but this way of defining it seems to me more difficult to show is bijective than the other.
Next i show that an element is a unit in one set if and only if it is a unit in the other set. This comes from the fact that (x, mn) = 1 if and only if (x, m) = 1 and (x, n) = 1 which is easy to prove. Therefore the number of units in both sets are the same and since the cardinality of the set is the value of the euler function the result follows.
i know i'm missing small details but are the contents of my proof correct? would it be beneficial to add or subtract anything in it? and as for defining a function that is bijective between the sets, was my TA right in cautioning me against defining my function in that way? thanks!