MHB You're welcome! Glad I could help. (Thumbs up)

[evinda]

Hi! (Wave)

Let $R$ be a relation.

Show the following sentences:

1. $dom(R^{-1})=rng(R)$
2. $rng(R^{-1})=dom(R)$
3. $fld(R^{-1})=fld(R)$
4. $(R^{-1})^{-1}=R$

That's what I have tried:

1. Let $x \in dom(R^{-1})$. Then $\exists y$ such that $<x,y> \in R^{-1} \Rightarrow <y,x> \in R \Rightarrow x \in rng(R)$.
   
   So, $dom(R^{-1}) \subset rng(R)$.
   
   Let $y \in rng(R)$. Then $\exists x$ such that $<x,y> \in R \Rightarrow <y,x> \in R^{-1} \Rightarrow y \in dom(R^{-1})$.
   
   So, $rng(R) \subset dom(R^{-1})$.
   
   Therefore, $rng(R)=dom(R^{-1})$.
   
2. Let $y \in rng(R^{-1})$. Then $\exists x$ such that $<x,y> \in R^{-1} \Rightarrow <y,x> \in R \Rightarrow y \in dom R$.
   
   So, $rng(R^{-1}) \subset dom R$
   
   Let $y \in dom R$. Then $\exists y$ such that $<x,y> ain R \Rightarrow <y,x> \in R^{-1} \Rightarrow x \in rng(R^{-1})$
   
   So, $dom R \subset rng(R^{-1})$.
   
   Therefore, $dom R = rng(R^{-1})$.
   
3. $$fld(R^{-1})=dom(R^{-1}) \cup rng (R^{-1})=rng(R) \cup dom R$$
   $$fld(R)=dom R \cup rng(R)$$

Is that what I have tried right or have I done something wrong? (Thinking)

Also, how could we prove the fourth identity $(R^{-1})^{-1}=R$ ?

 

[Evgeny.Makarov]

Number 1-3 are correct. Number 4 is even easier than 1 or 2.

 

[evinda]

Number 1-3 are correct. Number 4 is even easier than 1 or 2.

Is it maybe like that? (Thinking)

Let $<y,x> \in (R^{-1})^{-1}$. Then: $<x,y> \in R^{-1} \Rightarrow <y,x> \in R$.

So, $(R^{-1})^{-1} \subset R$.

Let $<x,y> \in R$. Then: $<y,x> \in R^{-1} \Rightarrow <x,y> \in (R^{-1})^{-1}$.

So, $R \subset (R^{-1})^{-1}$.

Therefore, $(R^{-1})^{-1}=R$.

(Thinking)

 

[Evgeny.Makarov]

Yes. I would write it as a chain of equivalences:
\[
\langle x,y\rangle\in R\iff \langle y,x\rangle\in R^{-1}\iff \langle x,y\rangle\in(R^{-1})^{-1}.
\]

 

[evinda]

Yes. I would write it as a chain of equivalences:
\[
\langle x,y\rangle\in R\iff \langle y,x\rangle\in R^{-1}\iff \langle x,y\rangle\in(R^{-1})^{-1}.
\]

Nice! I understand! (Nod) Thanks a lot! (Smile)