# 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

$R$ 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 $R$.
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