Question

Let t be an integer, and let n be a positive integer. Prove that
\[Z_n = \{[0],[1],.....,[n-1]\}=\{[t],[t+1],....,[t+(n-1)]\}\]
Please, help

Answer 1

I don't know what I am supposed to do. Is it not that t =0 ? set of \(Z_n\) are remainders in \(Z_n\) ?? so that they are equal iff t =0,

Answer 2

@ganeshie8

Answer 3

Notice that order doesn't matter for sets to be equal

Answer 4

But by definition, 2 sets are called equal iff each of their elements are equal.

Answer 5

I think you're trying to show that both the sets are equal in mod n

Answer 6

Oh, yes!! Thanks for the hint.

Answer 7

you need to show that the numbers : \(\large t, t+1, t+2, \ldots , t+(n-1)\) taken in some order give you all the residues mod n

Answer 8

is that correct ?

Answer 9

I am not sure. I have the left hand side has n distinct residue class modulo n. Now what I have to do is showing they are isomophic, right?

Answer 10

i think so,
take the right hand side and show that they form distinct residue class mod n

Answer 11

idk abstract algebra terminology, but we don't need a proof for that in number theory, we take it as obvious thing
let me see if i can cook up some proof

Answer 12

It's group theory.

Answer 13

And my prof uses "doctor up" not "cook up" hahaha...

Answer 14

suppose \(\large t+i \equiv t+j \pmod n\),
\(\large \implies i-j \equiv 0 \mod n \)
\(\large \implies n |(i-j) \) which is nonsense since \(\large i-j \lt n\)

That means all the elements in right hand side set are incongruent mod n to each other.
Since there are exactly \(\large n\) elements, they form a complete set of residues for mod n

Answer 15

Thanks for the proof. I do appreciate