In many mathematical books that I have read and from lectures from professors, the words 'into' and 'injective' were used interchangeably, but in Patrick Suppes book Axiomatic Set Theory he gives a percise definition of what it means for a function to be 'into':
$f$ is a function from $\:A$ into $\:B \leftrightarrow\: f$ is a function & $D(f) = A$ & $R(f) \subseteq B$
where $D(f)$ is the Domain of $f$ and $R(f)$ is the Range of $f$.
Is the definition given by Suppes the correct meaning of 'into', or is 'into' simply a synonym for 'injective'
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$\begingroup$Into is not a synonym for "injective". There is, however, another way of referring to an injective function: such a function is sometimes said to be "one-to-one function", which is not to be mistaken with a "one-to-one correspondence"/bijective function.
Even though we do refer to a surjective function as being "onto", it does not follow that an injective function is therefore "into."
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