Let R be a relation from the set A to the set B, then: Prove that Ran (R)=Dom (R-1 ).
Let RRR be a relation from the set AAA to the set BBB. Taking into account that R−1={(b,a) ∣ (a,b)∈R}⊂B×AR^{-1}=\{(b,a)\ |\ (a,b)\in R\}\subset B\times AR−1={(b,a) ∣ (a,b)∈R}⊂B×A, we conclude that Ran(R)={b ∣ (a,b)∈R}={b ∣ (b,a)∈R−1}=Dom(R−1).Ran (R)=\{b\ |\ (a,b)\in R\}=\{b\ |\ (b,a)\in R^{-1}\}=Dom (R^{-1} ).Ran(R)={b ∣ (a,b)∈R}={b ∣ (b,a)∈R−1}=Dom(R−1).
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