# into function vs injective function

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'

• I've always thought of "into" as basically telling you where the function's output lives; i.e., the codomain (or possibly range, if you want to emphasize that it's not the entire codomain). The term "one-to-one" generally means "injective". Jan 23, 2014 at 21:34
• That's makes more sense to me, so it would seem that Patrick's definition of 'into' is the proper definition Jan 23, 2014 at 21:36
• Can you give us a published (or on-line) example where "into" is used in this erroneous way? Suppes (and all other books I know) have the usual definition. May 30, 2015 at 15:21
• You can contrast "Into" meaning $R(f) \subseteq B$ with "onto" meaning $R(f) = B$. May 30, 2015 at 15:22

## 1 Answer

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."

• Well, according to Patrick Suppes it doesn't seem so. But so many people are using them as if they are. Just search into function in google Jan 23, 2014 at 21:33
• James, in that case, there are a lot of uninformed/ill-informed folks out there in Google Land. Jan 23, 2014 at 21:35
• +1 I like the last paragraph. It hadn't occurred to me that the use of the word "onto" certainly suggest a false relationship between "injective" and "into." Jan 23, 2014 at 21:36
• @amWhy No Doubt about that, but as I mentioned some creditable books have used the two terms interchangeably. Also, I really already knew the answer, but I just wanted to ask it so it would show up on google because other websites and forums give the wrong definition of 'into'. I will accept your answer. Thanks! Jan 23, 2014 at 21:37
• Very good strategy! Posting here $\implies$ creating a Google hit! Jan 23, 2014 at 21:42