# Equivalence relation and Equivalence classes?

• Jan 4th 2009, 05:04 AM
Jason Bourne
Equivalence relation and Equivalence classes?
I'm not sure on the following question:

(a) Define a relation R on Z by

$aRb$ if and only if $3|(a+2b)$

Show that R is an equivalence relation, and describe the equivalence classes.

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I'm particularly interested on how you prove the transativity part of the equivalence relation and how to describe the equivalence classes. Can anyone help?

(also any more explanation on Equivalence relations and classes would be much appreciated as I could certainly do with understanding it better.)
• Jan 4th 2009, 05:30 AM
Plato
Quote:

Originally Posted by Jason Bourne
(a) Define a relation R on Z by
$aRb$ if and only if $3|(a+2b)$
I'm particularly interested on how you prove the transativity part of the equivalence relation

$aRb\,\& \,bRc \Rightarrow \quad a + 2b = 3k\,\& \,b + 2c = 3j$

So add together: $\begin{gathered} a + 3b + 2c = 3k + 3j \hfill \\
a + 2c = 3k + 3j - 3b \hfill \\ \end{gathered}$
• Jan 5th 2009, 12:21 AM
Jason Bourne
thanks for that, anything on Equivalence classes?
• Jan 7th 2009, 03:39 AM
HallsofIvy
I would find symmetry most interesting because the rule defining equivalence does "look" symmetric! You must prove that if aRb, then bRa which means show that if a+ 2b is a multiple of 3, then so is b+ 2a. That is true but how did you show it?

An equivalence class consists of all those things that are equivalent to one another. Here two integers, a and b, are equivalent if and only if a= 2b is divisible by 3. Now just start looking at integers: If b= 0, then in order to be equivalent, a must satisfy "a+ 0= a is divisible by 3" so one equivalence class is just the multiples of 3. If b= 1, then we must have a+ 2 divisible by 3 which is the same as saying a is a multiple of 3 plus 1: 1, 4, 7, -2, -5, etc. If b= 2, then we must have a+ 4 divisible by 3: 2, 5, -1, -4, etc. Those are the only three equivalence classes: every integer is in one of those.