This is probably very naive but suppose I have an injective map from a class into a set, may I conclude that the domain of the map is a set as well?
Is the domain of an one-to-one function a set if the target is a set?
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2Perha$p$s you should clarify what is meant by "map" in this $q$uestion. The answers using replacement seem to be assuming that the map is defined by a formula in the language of set theory. – 2012-02-23
3 Answers
If a function $f:A\to B$ is injective one, we can assume without loss of generality that $f$ is surjective too (by passing to a subclass of $B$), therefore $f^{-1}:B\to A$ is also a bijection.
If $B$ is a set then every subclass of $B$ is a set, so $f^{-1}:B\to A$ is a bijection from a set, and by the axiom of replacement $A$ is a set.
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0The downvote seems a bit strange... – 2013-08-19
I say you can, since an injective map $f:A\to B$ is isomorphic to both its image (which is a set) and its domain.
Define $g$ on the range of $f$ such that $g(f(x)) = x$. This is well-defined because $f$ is injective. The domain of $g$ is equal to the range of $f$, which is a set.
Therefore by the axiom of replacement, or maybe the axiom of union applied to $\bigcup_{y\in{\rm Range}f} \{g(y)\} $, the range of $g$ is a set. But the range of $g$ is precisely the domain of $f$.
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0Thanks! I'm glad you like it. – 2014-10-15