B Onto set mapping is the surjective set mapping, and into injective?

[mcastillo356]

TL;DR Summary

I'm almost sure, but need to check it with the forum. It's about the first chapter of "Introduction to topology and modern analysis" I am reading online.

The textbook is being fine. I asked the forum for some introduction to topology, and decided to start with Simmon`s. This naive question is due to ignorance of the words into and onto, which I don't distinguish in Spanish. A quick browsing sugests I'm right.

 

[fresh_42]

Say, we have $f\, : \,A\longrightarrow B.$

$f$ is onto if it is surjective, i.e., $\forall\,b\in B\,\exists\,a\in A\, : \,b=f(a).$
$f$ is into if it is injective, i.e., $f(a)=f(a')\Longrightarrow a=a'.$

However, the English language is not always very exact when it comes to "in", "to", or "into". I have seen examples where into just meant $f(A)\subseteq B$ without the requirement of being injective. So, onto equals surjective is always true, into equals injective depends on the author. At least to my experience.

In German, we would use "auf" = "onto" = surjective, "in"="into"=injective, and "nach"="to"= $f(A)\subseteq B.$

 

[mcastillo356]

Say, we have $f\, : \,A\longrightarrow B.$

$f$ is onto if it is surjective, i.e., $\forall\,b\in B\,\exists\,a\in A\, : \,b=f(a).$
$f$ is into if it is injective, i.e., $f(a)=f(a')\Longrightarrow a=a'.$

Yes.

However, the English language is not always very exact when it comes to "in", "to", or "into". I have seen examples where into just meant $f(A)\subseteq B$ without the requirement of being injective. So, onto equals surjective is always true, into equals injective depends on the author. At least to my experience.

Simmons is being precise. Into means inyective.

In German, we would use "auf" = "onto" = surjective, "in"="into"=injective, and "nach"="to"= $f(A)\subseteq B.$

That's accurate, indeed.
Thanks, @fresh_42 !

 

[Gavran]

The textbook is being fine. I asked the forum for some introduction to topology, and decided to start with Simmon`s. This naive question is due to ignorance of the words into and onto, which I don't distinguish in Spanish. A quick browsing sugests I'm right.

To be on the set means occupying the whole set in a particular way and the word “onto” follows the definition of a surjective function at all.
To be in the set means occupying the part of the set but it does not mean that every element of the part of the set is occupied only once and this is the main reason for confusion. The word “into” does not completely follow the definition of an injective function.

 

[PeroK]

Whether you use into and/or onto is upto you!