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

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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 $Ran (R)=\{b\ |\ (a,b)\in R\}=\{b\ |\ (b,a)\in R^{-1}\}=Dom (R^{-1} ).$

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