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Define a relation $R$ on the set $A = \{n \mid n \in \mathbb{N} \textrm{ and } 0 < n < 14\}$ such that $R$ is an equivalence relation on $A$. (You can either define a property on $A$ or simply list the elements of $R$.) Then find the different equivalence classes of $R$ and show their connection to a partition of $A$.

I understand what $A$ is but I don't understand the rest of the question, can anyone point me in the right direction?

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I somewhat corrected the title. However I still think it is not a good title, but I cannot come up with a better title. –  Asaf Karagila Apr 22 '12 at 14:27
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1 Answer

up vote 3 down vote accepted

Recall that $R$ is a relation on $A$ if $R\subseteq A\times A$, that is the elements of $R$ are ordered pairs that both the elements come from $A$. If $\langle x,y\rangle\in R$ we often write $x R y$.

Let us define some properties of a relation $R$ on $A$:

  1. We say that $R$ is reflexive if for every $x\in A$ the pair $\langle x,x\rangle\in R$. That is $xRx$ is true for every $x\in A$.
  2. We say that $R$ is symmetric if for every $x,y\in A$ if $\langle x,y\rangle\in R$ then $\langle y,x\rangle\in R$. That is if $xRy$ then $yRx$.
  3. We say that $R$ is transitive if for every $x,y,z\in A$ if $\langle x,y\rangle,\langle y,z\rangle\in R$ then $\langle x,z\rangle\in R$. That is if $xRy$ and $yRz$ then $xRz$.

If $R$ has all three properties then $R$ is called an equivalence relation. Your question asks you to define an equivalence relation on $A$.

Next we suppose that $R$ is an equivalence relation on $A$, for $a\in A$ we define the equivalence class of $a$ (in $R$) to be the set: $$[a]_R=\{x\in A\mid aRx\}$$

Exercise: Prove that if $R$ is an equivalence relation on $A$ and $xRy$ then $[x]_R=[y]_R$.

The question now asks to take the equivalence relation which you have defined and write down all the equivalence classes.

Last you are require to discuss the relation between the equivalence classes and the characteristics of a partition of $A$. Let us define this as well:

We say that $P$ is a partition of $A$ if:

  1. Every $X\in P$ is a non-empty subset of $A$.
  2. For every $a\in A$ there is some $X\in P$ such that $a\in X$.
  3. For every $X,Y\in P$ if $X\neq Y$ then $X\cap Y=\varnothing$.
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@JamieB Also, even if $R = id$ and $R = A \times A$ are correct choices, I suspect that the author of the original question wanted you to pick something non-trivial. –  dtldarek Apr 22 '12 at 14:34
    
@dtldarek: Did you post this comment here to make sure whoever reads it reads it after reviewing (or better yet: reading) my answer? –  Asaf Karagila Apr 22 '12 at 14:36
    
Yes, exactly. As you have written almost everything I would and even more, it seemed that the sensible action would be to add that as a comment ;-) –  dtldarek Apr 22 '12 at 14:55
    
When I get to the required reputation I'll accept this as the answer, many thanks :) –  JamieB Apr 22 '12 at 15:17
    
@JamieB: I thought that you can always accept an answer, you just need reputation for upvoting. Either way you should have about 10 more points which should be about enough to do the basic things on the site. –  Asaf Karagila Apr 22 '12 at 15:18
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