# Sets and Relations

I've been having quite a bit of trouble with this topic and I'm in need of some help with this question.

Determine whether easy relation is reflexive, symmetric, or transitive.

$A=\mathbb{R}$; $(a,b)$ is an element of $R$ if and only if $a - b \le 3$.

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I'm not sure what you're struggling on. Have you worked out whether any of the three properties hold? Two should be very easy, provided you understand the definitions. One might take a bit more thought. – Billy Sep 30 '11 at 7:35

## 1 Answer

This is mostly just an exercise in reading and applying the definitions of reflexivity, symmetry, and transitivity. I’ll get you pointed in the right direction, but you’ll have to do most of the work.

$R$ is reflexive if $(a,a)\in R$ for every real number $a$; is this true? That is, is it true that $a-a\le 3$ for every real number $a$?

$R$ is symmetric if $(b,a)\in R$ whenever $(a,b)\in R$; is this true, or are there real numbers $a$ and $b$ such that $(a,b)\in R$ but $(b,a)\notin R$? Begin by rewriting that question in terms of this specific relation: are there real numbers $a$ and $b$ such that $a-b\le 3$ but $b-a\not\le 3$?

$R$ is transitive if $(a,c)\in R$ whenever $(a,b)\in R$ and $(b,c)\in R$. Can you find three real numbers $a,b$, and $c$ such that $(a,b)\in R$ and $(b,c)\in R$, but $(a,c)\notin r$? Begin by translating ‘$(a,b)\in R$’ and so on into inequalities, using the definition of $R$.

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So the relation would definitely be reflexive and symmetric from what I've read? The transitive part is still a bit fuzzy though. – BleuCheese Sep 30 '11 at 8:35
It is reflexive. For symmetry, take a look at $a=0$ and $b=4$, for instance. For transitivity, try $a=4$, $b=2$, and a variety of values for $c$. – Brian M. Scott Sep 30 '11 at 8:38
This makes perfect sense now. I was limiting myself to certain numbers and not exploring the full range. Thank you so much. – BleuCheese Sep 30 '11 at 8:41