Dawson Downs

2022-07-16

Let R be a relation on E. Demonstrate that:
- R is reflexive if and only if $I{d}_{E}\subseteq R$;
- R is symmetric only and only if $R=R-1$.

Abraham Norris

Expert

Step 1
Writing this kind of proof is mostly remembering the definitions. suppose R is a relation over A.for the first point, use the definition:
$\text{R is reflexive}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }x\in A.\in R$
and we have:

Step 2
for the second one, we use the definition:
$\text{R is symmetric}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }\in R.\in R$
and so we get:
$\text{R is symmetric}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }\in R.\in R\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\left\{|\in R\right\}\subseteq R\phantom{\rule{0ex}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}{R}^{-1}\subseteq R$

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