Ryuzaki
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Homework Statement
Prove that the relation, two finite sets are equivalent if there is a one-to-one correspondence between them, is an equivalence relation on the collection S of all finite sets.
I'm sure I know the gist of how to do it, but I'm a beginner in proofs, and I'm not sure if I've written it down correctly. I absolutely encourage nitpicking in the following proof, as I wish to learn how proofs are correctly written. Thanks!

Homework Equations
N/A
The Attempt at a Solution
Let S be the class of all finite sets.
Let A, B and C be three finite sets.
Reflexive property
Now, n(A) = n(A), and hence there exists a one-to-one correspondence between A and A
Therefore, A \approx A ------------------(1)
Symmetric property
Let A \approx B
\Rightarrow n(A) = n(B)
\Rightarrow n(B) = n(A), and hence there exists a one-to-one correspondence between B and A
\Rightarrow B \approx A
Therefore, A \approx B \Rightarrow B \approx A----------------(2)
Transitive property
Let A \approx B
\Rightarrow n(A) = n(B)---------------------(3)
Also, let B \approx C
\Rightarrow n(B) = n(C)---------------------(4)
From (3) and (4), n(A) = n(C)
\Rightarrow A \approx C
Therefore, A \approx B and B \approx C \Rightarrow A \approx C--------(5)
From (1), (2) and (5), it is clear that the relation, two finite sets are equivalent if there is a one-to-one correspondence between them, is an equivalence relation.
Q.E.D