Let \( f:A\to B \) with \( A=\{0,1,2\} \) and \( B=\{a,b\} \). How many injective functions from \( A \) to \( B \) exist, and which statement about replacement is correct?
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D. Zero; and replacement guarantees that the image of \( A \) under a formula is a set.
There are no injections from a three-element set into a two-element set (pigeonhole principle), so the count is zero. Replacement ensures the image of a set under a formula is a set.
The Zermelo-Fraenkel axioms (ZF) are the standard axiom system of set theory, formulated in first-order logic. They allow the usual mathematics to be built from the primitive notions of set and membership (\( \in \)).
\[\forall x\,\forall y\,(\forall z\,(z\in x \leftrightarrow z\in y)\rightarrow x=y)\]