Thread: Total relation

How can I proof that if $\displaystyle L$ is a universal relation that $\displaystyle R total relation\equiv L = L \circ R$. It is pretty obvious that it is true, but how do I prove it?

Could you explain what Rtotalrelation is? Also, is universal relation the one that contains all possible pairs? Finally, by $\displaystyle L\cdot R$ do you mean the composition $\displaystyle L\circ R$?

Originally Posted by emakarov

Could you explain what Rtotalrelation is? Also, is universal relation the one that contains all possible pairs? Finally, by $\displaystyle L\cdot R$ do you mean the composition $\displaystyle L\circ R$?

R total relation is

$\displaystyle R \equiv (\forall x |\exists z|: xRz))$

Yes, the universal relation is the one that contains all pairs

Yes, I did mean the composition. Edited the OP.

This is just a matter of using the notation.
If $\displaystyle (a,b)$ is any pair then $\displaystyle \left( {\exists c} \right)\left[ {(c,b) \in R} \right]$ because $\displaystyle R$ is total.
Because $\displaystyle L$ is universal $\displaystyle (a,c)\in L$ so $\displaystyle (a,b)\in L\circ R.$

Originally Posted by Plato

If $\displaystyle (a,b)$ is any pair then $\displaystyle \left( {\exists c} \right)\left[ {(c,b) \in R} \right]$ because $\displaystyle R$ is total.
Because $\displaystyle L$ is universal $\displaystyle (a,c)\in L$ so $\displaystyle (a,b)\in L\circ R.$

I think it should say that for any (a,b) there exists a $\displaystyle c$ such that $\displaystyle (a,c)\in R$ because R is total. Then $\displaystyle (c,b)\in L$ because L is universal, so $\displaystyle (a,b)\in L\circ R$.

Thanks, that is correct. It was either to early or the cut and paste.