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Take the relation $R$ to be defined on the set of integers:

$$aRb \iff 5 \mid (a + 4b)$$

As part of a larger proof, I have to show that $R$ is both symmetric and transitive. I'm lost.

I see the first steps, but I can't find how to progress further. Here's what I have at this point:

Proof of Symmetry

We have to prove that if $5 \mid (a + 4b)$, then $5 \mid (b + 4a)$. Clearly, this is true if $a = b$, but apart from that, it's unclear in my mind.

Proof of Transitivity

We have to prove that if $5 \mid (a + 4x)$ and $5 \mid (x + 4b)$, then $5 \mid (a + 4b)$.

I've fiddled around with sample values, but I still don't see it. I'm pretty lost here. Thoughts?

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$aRb$ iff $5\mid a-b$ iff $5\mid b-a$. – Did Dec 4 '11 at 22:34
a hint: $(a+4b) + (b+4a) = (5a + 5b)$ – deinst Dec 4 '11 at 22:36
I've changed "Symmetric proof" to "Proof of Symmetry". "Symmetric proof" reads as if you are saying that the proof is (somehow) symmetric, rather than that the proof is a proof about symmetry. "Symmetry proof" would have been okay, too... – Arturo Magidin Dec 4 '11 at 23:28
up vote 4 down vote accepted

HINT $\rm\ \ a\:R\:b\ \iff\ a-b\in 5\:\mathbb Z\:.\:$ If so, negating yields that $\rm\: b-a\in 5\:\mathbb Z\: $ hence $\rm\:R\:$ is symmetric. Transitivity follows by addition: $\rm \ a-b\:,\ b-c\:\in 5\:\mathbb Z\ \Rightarrow\ a-b + b-c\ =\ a-c \in 5\:\mathbb Z\:.$

Hence symmetry arises from $\rm\:5\:\mathbb Z\:$ being closed under negation, and transitivity arises from $\rm\:5\:\mathbb Z\:$ being closed under addition, i.e. from it being an additive subgroup of $\rm\mathbb Z\:.$ The innate algebraic structure will be brought to the fore when you study congruences and ideals of rings.

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Symmetry: If $5\mid a+4b$, does $5\mid \big((a+4b)-(5a+5b)\big)$?

Transitivity: If $5\mid a+4x$ and $5\mid x+4b$, what can you say about $(a+4x)+(x+4b)$? How is this number related to $a+4b$?

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@deinst's comment cleared up the symmetry part for me, but I don't see why adding $a+4x$ to $x+4b$ helps us solve the transitivity part of the proof? – David Chouinard Dec 4 '11 at 22:52
$(a+4x)+(x+4b)=(a+4b)+5x$, so $a+4b=(a+4x)+(x+4b)-5x$. What do you know about each of the three terms on the righthand side? – Brian M. Scott Dec 4 '11 at 22:59

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