Does anyone know if these following two statements can be proved somehow or are they simply true by definition?
"If there exists a surjective function , then "
"If there exists an injective function , then "
Both statements are invoked in more advanced proofs without hesitation, so I usually take them for granted. But I realized that's a bad mentality, so I tried to prove those two statements using the definition of injection and surjection but I am not sure if my approach was correct.
For a surjection, it specifies that for all , there exists an such that . The definition never specified that such an was unique, so we could construct a mapping of two elements or more elements in towards one element in , which would get us a comparison of the cardinality of the two sets. Is that correct (and if it is, rigorous enough)?
For an injection, it only says that if two elements , then . It never talked about such an element which has no corresponding element to map from, which would suggest that could be less than . Would that work?
I could not find anything online that delved further into the matter. Could someone please shed some light on this matter?