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I am confused by the question of how to define surjectivity for multivalued mapping( if this definition has already existed).

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We say that a function $f: A \rightarrow B$ is surjective if for all $b \in B$ you can find an $a \in A$ such that $f(a)=b$. If you are working with a multivalued mapping (i.e. if the relation $f$ is not necesarily univoque), I think that the notion of surjectivity will be the same that the one I stated above.

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