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We are doing some more with relations and this time we are given a relation and told that it is not equivalent. We need to find out which property it does not fulfill. So we at least know there is going to be one, maybe more.

Here is the relation: $aRb$ iff either $a$ mod 4 = $b$ mod 4 or $a$ mod 6 = $b$ mod 6, over $\mathbb{N}$.

So I assume we are supposed to show whether or not its reflexive, symmetric, and transitive?

Reflexive: Suppose $x \in \mathbb{N}$. Then, either $x$ mod 4 = $x$ mod 4 or $x$ mod 6 = $x$ mod 6. Either way, this shows that $xRx$, therefore the relation is reflexive.

Symmetric: Here is where I think I am getting confused. Suppose $x, y \in \mathbb{N}$. Then $x$ mod 4 = $y$ mod 4 and $y$ mod 4 = $x$ mod 4. The same thing applies for $x$ mod 6 = $y$ mod 6. Therefore, the relation is symmetric? (Is that right?)

Transitive: If $xRy$ and $yRz$ then, $xRz$. So, if $x$ mod 4 = $y$ mod 4 and $y$ mod 4 = $z$ mod 4, then would $x$ mod 4 = $z$ mod 4? If so, then this property is transitive as well.

So to me, I am getting that is an equivalence relation since I got it to be reflexive, symmetric, and transitive. So I know I am doing something wrong since the directions clearly state that it is not one. I am not sure on the symmetric portion. I think that's where I am messing up. Any help?

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But for the transitive part, remember what your relation says. It says that $a\pmod 4 = b\pmod 4$ OR $a\pmod 6 = b\pmod 6$. So if $xRy$ and $yRz$, you can have that $x\pmod 4 = y\pmod 4$ and that $y\pmod 6 = z\pmod 6$. Do you then have that $xRz$?

I hope this clears it up, otherwise let me know.

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  • $\begingroup$ It says a mod 4 = b mod 4. I don't know if that changes it since it doesn't just equal b, it equals b mod 4 or b mod 6. Would that have any affect on it? I do see that if a mod 4 or a mod 6 both equaled b by itself then it would not be transitive. $\endgroup$ – generic user007 Apr 28 '15 at 19:31
  • $\begingroup$ Ok, let me correct my answer. The point will be the same, however. Can you find numbers $a$, $b$, and $c$, such that $a\pmod 4 = b\pmod 4$, and $b\pmod 6 = c\pmod 6$, but $a$ is not related to $c$? $\endgroup$ – Mankind Apr 28 '15 at 19:33
  • $\begingroup$ Oh, okay. I see what you're saying now. That makes a lot more sense. Thank you :) Does the rest of it look okay though? Like the reflexive and symmertirc? $\endgroup$ – generic user007 Apr 28 '15 at 19:37
  • $\begingroup$ Good. :) Yeah, the rest looks good. $\endgroup$ – Mankind Apr 28 '15 at 19:47

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