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 $a - b \le 3$.
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 $R$.