1. ## Equivalence Relations

I'm stuck with this equivelance relations question,

'Let ~be the relation on the set Z of all integers defined by m~n provided that 10m+ n is divisible by 11.

1. Prove that ~ is an equivalence relation.'

First for reflexitivity, we do m~m and thus 10m+m=11m which surely is divisible by 11.

Now for it being symmetric, we do n~m which means 10n+m and we need to prove that that is divisible by 11, however do not not how to prove this.

Finaly for transitivity, I do not know how to approach it here - its kinda got me confused.

Help would be very much appreciated, thank you very much.

2. ## fixed typo

Originally Posted by iPod
I'm stuck with this equivelance relations question,
'Let ~be the relation on the set Z of all integers defined by m~n provided that 10m+ n is divisible by 11.
1. Prove that ~ is an equivalence relation.'
You see that m~n tells us that 10m+n=11k for some k.
Can you see see that 10n+m=110k-99m?
If so, is n~m?

3. I'm sorry, but I wasn't able to get to your answer, all I have managed to do is;
$\displaystyle 10n+m=(10m+n)+9n-9m$
$\displaystyle 10n+m=11k+9(n-m)$

could you hint me out how you got 10n+m=11(10k-9)??

4. $\displaystyle 10m+n=11k$

$\displaystyle 100m+10n=110k$

$\displaystyle 10n+m=110k-99m$

5. Originally Posted by iPod
I'm stuck with this equivelance relations question,

'Let ~be the relation on the set Z of all integers defined by m~n provided that 10m+ n is divisible by 11.

1. Prove that ~ is an equivalence relation.'

First for reflexitivity, we do m~m and thus 10m+m=11m which surely is divisible by 11.

Now for it being symmetric, we do n~m which means 10n+m and we need to prove that that is divisible by 11, however do not not how to prove this.

Finaly for transitivity, I do not know how to approach it here - its kinda got me confused.

Help would be very much appreciated, thank you very much.

Plato gave you the hint for symmetry. For transitivity, you want to show that if m~n and n~t, then m~t.

m~n and n~t means that 10m + n = 11k and 10n + t = 11r for some integers k and r. In light of these two equations, what can we say about the expression 10m + t?

6. I have managed to get as far as 100m-t=11(k-r) but i cant get it in the form of 10m+t,

is there a specific technique i could follow, because ive been trying for quite a long time and i concluded that there may be a technique that i am not aware of.

7. $\displaystyle 10m+n=11k\text{ and }10n+t=11j$

$\displaystyle 10m+11n+t=11k+11j$

So what in next?

8. I would like to point out that i just done the exact same method now and i got the answer lool :P
10m+t=11(k+j-n)

but this has actually helped expand my knowldge for answering questions in the future, thank you very much