Answer to Question #179903 in Discrete Mathematics for Johnson

Question #179903

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


1
Expert's answer
2021-04-13T13:45:53-0400

Let RR be a relation from the set AA to the set BB. Taking into account that R1={(b,a)  (a,b)R}B×AR^{-1}=\{(b,a)\ |\ (a,b)\in R\}\subset B\times A, we conclude that Ran(R)={b  (a,b)R}={b  (b,a)R1}=Dom(R1).Ran (R)=\{b\ |\ (a,b)\in R\}=\{b\ |\ (b,a)\in R^{-1}\}=Dom (R^{-1} ).


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