did he state that $i,j \in \mathbb{Z}_n$ ?
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?