Proving "[a]_n ∩ [b]_n is Empty or Equals [b]_n

[kathrynag]

Homework Statement

Prove that either $$[a]_{n}$$$$\cap$$$$<b>_{n}</b>$$=empty set or $$[a]_{n}$$=$$<b>_{n}</b>$$.

Homework Equations

The Attempt at a Solution

I want to assume there is an element x in $$[a]_{n}$$$$\cap$$$$<b>_{n}</b>$$ and show this implies $$[a]_{n}$$=$$<b>_{n}</b>$$.
This tells me x is in $$[a]_{n}$$ and $$<b>_{n}</b>$$.
That's where I get stuck.

 

[kathrynag]

sorry, I meant to show this implies [a]=.

 

[Office_Shredder]

If x=a mod n then n|x-a. x=b mod n means n|x-b.

If x|x-a and n|x-b... what can you say about a-b?

 

[kathrynag]

x-a=x-b
x-x=a-b
0=a-b
b=a

 

[Office_Shredder]

x-a is probably not equal to x-b

 

[kathrynag]

Ok then I'm not really sure where to go with a-b then.