We already know that the definition of one to one correspondence between set A and B
is as follow:
Every element in A
corresponds to exactly one element to B
Every element in B
corresponds to exactly one element to A
Now, the question is that what does corresponds mean, is this a two way relationship?
Two prove they are one to one corresponds, we need to show 1. and 2.
For element n∈A,
the corresponding element in B is n+1
For element n∈B,
the corresponding element in A is n−1
and we are done.
The question is can I have another correspondence in 2. (only in 2, not 1). Say 51 -1, 50-2, 49-3, etc. Then we also have that Every element in B
corresponds to exactly one element to A. However, the correspondence are different. Is this valid? If not, when you are proving a one to one correspondence, do you need to show the correspondence is consistent in both 1. and 2.? I think this question is very much related to the definition of correspondence, so I asked this question above.