Question

Let R be a relation from the set A to the set B, then: Prove that Ran (R)=Dom (R-1 ). 

Answer 1

Let $R$ be a relation from the set $A$ to the set $B$. Taking into account that $R^{-1}=\{(b,a)\ |\ (a,b)\in R\}\subset B\times A$, we conclude that $Ran (R)=\{b\ |\ (a,b)\in R\}=\{b\ |\ (b,a)\in R^{-1}\}=Dom (R^{-1} ).$