I'm trying to make sense, well better sense, of a proof of the following that my instructor gave me, while I work out my own proof of the same if then statement.

The problem is;

Let and . Consider the set

Prove that every integer is congruent modulo n to some element of S.

The given proof is;

He introduces the definition of cardinality for a set and notes that the cardinality of equals the cardinality of

(1) Let

(2) Solve

(3) Show that

(4)

(5) and

So the set S is equivalent to the set We have a Theorem that states, every integer is congruent modulo n to an element of H.

end proof.

So the part about the cardinality of S and H makes perfect sense. Since all S is is H with a k added to every element.

From 2 to 3 looks good to me. He just set

Proving the bi-conditional in 3 is giving me a problem.

Is trivial.

The other direction is giving me some trouble.

Assume

and

I don't get how this shows that i and j are equal. Does it show that they are they additive inverses of each other?