Mapping/Quantifier/Interpretation as solution/Remark

The question, whether a mapping ${}F\colon L\rightarrow M$ has the properties of being injective or surjective, can be understood looking at the equation

${}F(x)=y\,$

(in the two variables ${}x$ and ${}y$). The surjectivity means that for every ${}y\in M$ there exists at least one solution

${}x\in L\,$

for this equation, the injectivity means that for every ${}y\in M$ there exist at most one solution ${}x\in L$ for this equation, and the bijectivity means that for every ${}y\in M$ there exists exactly one solution ${}x\in L$ for this equation. Hence surjectivity means the existence of solutions, injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics and they also can be interpreted as surjectivity or injectivity of suitable mappings.