Equivalence Relation Question

[connor_cee]

Was doing a paper and came across an question about equivalence relations:

Let U = {-3, -3, -1, 0 , 1, 2, 3... 16}

and let R be the relations between two elements of U such that aRb if a = b mod 4

the question then goes onto say show that R is an equivalence relation,
any ideas?

 

[pka]

Was doing a paper and came across an question about equivalence relations:
Let U = {-3, -3, -1, 0 , 1, 2, 3... 16}
and let R be the relations between two elements of U such that aRb if a = b mod 4
the question then goes onto say show that R is an equivalence relation,
any ideas?

First I am sure you mean \(\displaystyle U=\{-3,-2,-1,0,\cdots,16\}\).
The relation \(\displaystyle a\mathcal{R}b\) if and only if \(\displaystyle (b-a)\mod 4=0\)

One equivalence class is: \(\displaystyle \mathcal{R}/-3=\{-3,1,5,9,13\}\)

Now you find all the other equivalence classes.
Prove that they partition \(\displaystyle U\).

 

[connor_cee]

First I am sure you mean \(\displaystyle U=\{-3,-2,-1,0,\cdots,16\}\).
The relation \(\displaystyle a\mathcal{R}b\) if and only if \(\displaystyle (b-a)\mod 4=0\)

One equivalence class is: \(\displaystyle \mathcal{R}/-3=\{-3,1,5,9,13\}\)

Now you find all the other equivalence classes.
Prove that they partition \(\displaystyle U\).

I copied the question straight from the past paper that I was doing, so what you see is what was on the paper, your copy was correct except for the \(\displaystyle (b-a) \) part that wasn't written in the paper. I am confused because of the answer the professor wrote, which was and I apologize for not using the mathematical format I'm new to this forum:


a = b mod 4 i.e. a - b = 4k , k is an integer

a - a = 4 x 0, so aRa[/tex]

a - b = 4k, so b - a = -4k i.e. aRb => bRa

a - b = 4k , b - c = 4L so a - c = (a-b) + (b-c) = 4(k + L) so if aRb and bRc then aRc



This is the answer the professor wrote, and I know there are three conditions needed to show a equivalence relation (symmetric, transitive, reflexive) , what i don't understand is how, with this relation, how he shows it algebraically, probably my weakness in modular arithmetic, if someone could explain hes part that would be amazing.

 

[pka]

This is the answer the professor wrote, and I know there are three conditions needed to show a equivalence relation (symmetric, transitive, reflexive) , what i don't understand is how, with this relation, how he shows it algebraically, probably my weakness in modular arithmetic, if someone could explain hes part that would be amazing.

I wish you had said that to begin with because then I would not have posted an answer. I simply refuse to explain someone else's proof/solution. In general, most of us do thing differently from one another.