“$f$ 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?

• some people prefer "onto" instead of "surjective", and then "into" makes sense as the analogue of "injective", but i think most people would just interpret it as the same as "$f$ is a function from $A$ to $B$" – citedcorpse Jun 14 '13 at 17:34
• I think this is a perfectly fair question, unsure why it was downvoted. – icurays1 Jun 14 '13 at 17:36
• As a non-english motherboard I'd kill anyone that uses this kind of terminology, since into, onto, to seem more or less the same and are easily confused. I'd personally say "$f$ is an injection/surjection from $A$ to $B$" if I want to add that information without adding adjectives. – Bakuriu Jun 14 '13 at 21:10

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 $B$.)

• Nice write up! +1 – Amzoti Jun 15 '13 at 1:16
• Thank you for the edit, @ruakh! – Namaste Jun 15 '13 at 2:11
• As Celine songs: Fly, Fly little one....:-) – mrs Jun 15 '13 at 5:06
• @amWhy , thanx , one strange thing is that i have 17+ upvotes ! when i asked the question , i hoped that it will not be closed because it doesn't satisfies the politics of the site as it has no useful information for others in future ! – Fawzy Hegab Jun 15 '13 at 13:40
• Thanks, MathsLover! I love math as much as you ;-) – Namaste Jun 16 '13 at 17:56

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

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.

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"). 