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Onto vs into

  1. Apr 29, 2010 #1
    "Onto" vs "into"

    What is the difference between a homomorphism which is from a group G ONTO a group H and a homomorphism which is from a group G INTO a group H?
     
  2. jcsd
  3. Apr 30, 2010 #2

    radou

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    Re: "Onto" vs "into"

    "Into H" only means that the codomain of the mapping is H, and "onto H" means that the codomain equals the image of the mapping, i.e. it is surjective.
     
  4. Apr 30, 2010 #3

    Hurkyl

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    Re: "Onto" vs "into"

    Sometimes, into is used to imply injective.
     
  5. Apr 30, 2010 #4

    Landau

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    Re: "Onto" vs "into"

    Can you give an example / reference? I have never come across this meaning.
     
  6. Apr 30, 2010 #5
    Re: "Onto" vs "into"

    Same, all the books I have seen in Algebra use onto when the function is surjective, otherwise they say into.
     
  7. Apr 30, 2010 #6

    Fredrik

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    Re: "Onto" vs "into"

    It should also be mentioned that "into" doesn't imply that the function isn't surjective. We can definitely talk about a surjection from X into Y. "Into" is the word you use by default, and you can change it to "onto" if you're allergic to French or something*, so that you need to say that the function is surjective without actually using that word.

    *) I don't know if "surjective" should be considered a French word, but I read that the term was introduced by Nicolas Bourbaki.
     
  8. Apr 30, 2010 #7
    Re: "Onto" vs "into"

    "Into" is a meaningless word.

    "Onto", on the other hand, is a synonym for "surjective". (Sur = on in French, that's how I remember it). A surjection is a functions where the range equals the codomain.

    Surjections are usually studied together with injections (often called "one-to-one" functions). A function is surjective and injective, we call it bijective (or "one-to-one onto").
     
  9. Apr 30, 2010 #8
    Re: "Onto" vs "into"

    Wot?
     
  10. Apr 30, 2010 #9

    D H

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    Re: "Onto" vs "into"

    That is exactly that I was taught, years and years (decades and decades) ago. For those of us who were allergic to French, into=injective, onto=surjective, into and onto=bijective.
     
  11. Apr 30, 2010 #10
    Re: "Onto" vs "into"

    I think you will get into trouble nowadays if you assume "into" means injective.
     
  12. Apr 30, 2010 #11
    Re: "Onto" vs "into"

    I bet my father's older than your father, but neither into nor onto meant 1-1 when I was at school!
     
  13. Apr 30, 2010 #12

    lavinia

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    Re: "Onto" vs "into"

    into means injective - when you say a mapping is into. It means nothing if you say a map takes a space into another.
     
  14. May 1, 2010 #13
    Re: "Onto" vs "into"

    My very limited experience matches what Radou and Fredrik have said, e.g.

    Carol Whitehead: Guide to Abstract Algebra:

    It's only later that she introduces the concept of injective (one-to-one) mappings.

    On the other hand, Borowski & Borwein: Collins Dictionary of Mathematics:

    On it's own, given their own definition of "into", the final comment is confusing to me. But the entry INJECTIVE says that, according to one usage, one-to-one means injective (this is the usage I'm familiar with), whereas, according to another usage, one-to-one means bijective. So maybe that final comment is talking about the one-to-one = bijective usage, in which case maybe they're saying that, according to that usage, into means injective.

    I reckon it's a consipracy to keep us all using French, allergic or not ;-)
     
  15. May 1, 2010 #14
    Re: "Onto" vs "into"

    I can't understand why people keep saying "into" is meaningless. Why doesn't it mean into? What exactly is supposed to make it meaningless compared with the other terms describing mappings?

    I'm mystified. Can someone give me a clue.
     
  16. May 1, 2010 #15

    lavinia

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    Re: "Onto" vs "into"

    into is used in two ways.

    - A maps takes one space into another. Here it tells you nothing about the map.

    - A map is into. Here into means that the map is injective.
     
  17. May 1, 2010 #16
    Re: "Onto" vs "into"

    So you presumably mean it's ambiguous to the point of being not worth using?

    Doesn't the first statement imply that none of the images of the map fall outside the second space? This is the usage I am used to. The books I have that use the English descriptions seem to consistently use it to mean exactly that, though they also tend to use "to" and "into" interchangeably.

    By "injective" I would understand 1-1. Is that also your understanding? I am another one who has never seen "into" used in this way. Where is this use from?
     
  18. May 1, 2010 #17

    D H

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    Re: "Onto" vs "into"

    No, it appears to me that lavinia is saying the word has multiple meanings. That does not mean its ambiguous. The meaning is usually rather clear from context and usage.

    For a good example of a word (or phrase) that is closer being too ambiguous is 1-1. You are interpreting that as meaning injective. Others use it to mean bijective.
     
  19. May 2, 2010 #18

    lavinia

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    Re: "Onto" vs "into"

    The use of into as injective is standard. You should expect to see it used that way.
     
  20. May 2, 2010 #19

    Landau

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    Re: "Onto" vs "into"

    Please, give references for 'into=injective' being standard. I don't recall having read a book which uses this terminology.
     
  21. May 2, 2010 #20

    D H

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    Re: "Onto" vs "into"

    Stop quibbling over semantics. Several members have noted that they have either been taught or know that others have been taught that into can be a synonym for injective.

    In the first meaning into is used as a preposition. In the second it is used as an adverb. There is *no* ambiguity here.

    One-to-one on the other hand is ambiguous. To some authors one-to-one is a synonym for injective while for others it is a synonym for bijective. A reader who comes across that term in a text or a paper had either know what that particular author means by that term or infer the meaning from its usage.

    Just because a word has multiple meanings does not mean that the word is useless. Here is a graph:

    Cubicpoly.png

    Here is another graph:

    250px-6n-graf.svg.png

    Does these multiple meanings mean we should avoid using the word "graph"? Of course not.
     
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