Just introduced to discrete math and I am stuck with following demonstrations.
Let $ R $ be a relation on $ E $. Demonstrate that:
- $ R $ is reflexive if and only if $ \operatorname {Id} _ E \subseteq R $;
- $ R $ is symmetric only and only if $ R = R ^ { - 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: $$\text{R is reflexive} \iff \forall x\in A.<x,x>\in R$$
and we have: $$R \text{ is reflexive} \iff \forall x\in A.<x,x>\in R \iff \{<x,x>|x \in A \} \subseteq R \iff Id_A \subseteq R $$ $\blacksquare$
for the second one, we use the definition: $$ \text{R is symmetric} \iff \forall <x,y>\in R. <y,x>\in R$$ and so we get: $$\text{R is symmetric} \iff \forall <x,y>\in R. <y,x>\in R \iff \{<y,x>|<x,y>\in R\} \subseteq R \\ \iff R^{-1} \subseteq R$$
$\blacksquare$
hope this helps.