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# How to prove a one to one correspondence.

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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?

Example:

Given A=1,2,3,4,5,⋯,50

and B=2,3,4,5,6,⋯,51

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.

Oct 23, 2021