“$f$ is a function from $A$ to $B$” vs. “$f $is a function from $A$ into $B$”?

When we say that

$f$ is a function from $A$ to $B$

is this different from saying

$f$ is a function from $A$ into $B$

I know what injective ("1-1"), surjective ("onto"), and bijective functions are, but is there such a thing as an "into" function?

Solution 1:

Both expressions say the same thing.

But note that saying "$f: A \to B$ is a function from $A$ into $B$ does not imply that $f$ is into but not onto (i.e., it does not rule out that $f$ might be onto, so it is not the "converse" of, or the negation of, the descriptor "onto" or "surjective"). Indeed, it says nothing more, and nothing less, than the alternative: "$f$ is a function from $A$ to $B$."

• So we can either say that there is no such thing as an "into function" (i.e. it is NOT used to describe some, but not all functions, nor is it describe particular kinds of functions),

• Or we can say that every function is into (meaning every function $f: A \to B$ is a function on $A$ into $B$.)

Solution 2:

Both mean the same thing. There is no such thing as an "into" function.

Solution 3:

Any function $f : A \to B$ is said to map $A$ into $B$ or to be a mapping from $A$ into $B$. The term into is used in general for any function, it doesn't relate to any specific kind of functions.

Solution 4:

There is no difference between "to" and "into" here.

In Gelbaum and Olmstead's Counterexamples in Analysis, they use this terminology in a way that makes it seem slightly more reasonable (though I personally still wouldn't ever use "into").