A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

plearse teach me injective, surjective and byjective

  • This Question is Closed
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @zzr0ck3r

  2. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Do you know what a function is?

  3. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    you are given variables and replaced by numbers for short

  5. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    A function is one-to-one (injective) if every x in the domain maps to at most one y in the codomain example |dw:1441760363692:dw| here is a non example |dw:1441760400010:dw| Does this make sense?

  6. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no. please explain

  7. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    notice that in the first picture each element in the domain (the one on the left) gets sent to only one element on the right (this is the codomain) Notice this is NOT the case in the non example

  8. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok. so the first is injective?

  9. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    correct

  10. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    the second one is not because 1 and 2 both get sent to a

  11. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  12. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    in any function, every x maps to at most one y in the range

  13. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    A function is onto (surjective) if everything in the codomain gets used up example |dw:1441760739646:dw| non example |dw:1441760788586:dw|

  14. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    read this http://www.math.ucla.edu/~tao/java/MultipleChoice/functions.txt

  15. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  16. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    " For every x in X there is at most one y in Y such that f(x) = y." Comment. Every function f has this property (they each map one element to one element, i.e. they are not "one-to-two"). However, this is not what one-to-one means.

  17. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    A function must have 2 properties 1) it is defined everywhere i.e. everything in the domain gets used up notice how surjectivity is sort of like the opposite of this 2) it is well defined i.e. each x can map to at most one y notice how injectivity is the opposite of this

  18. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Ok @GIL.ojei Is the following function surjective, injective both, or neither? |dw:1441761073116:dw| ?

  19. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    injective

  20. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    why?

  21. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    @jayzdd I think it is much better to start with a intuitive notion. this notation is not going to help imo. that comes next

  22. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    a better definition for injectivity distinct (different) inputs map to distinct outputs formally: if a ≠ b, then f(a) ≠ f(b) this is equivalent to if f(a) = f(b), then a = b

  23. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    @GIL.ojei why?

  24. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes you're diagrams are good to explain the intuition. i was taking issue with the phrase 'every x has at most one y ' for your one to one thats true about all functions

  25. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    because each maps differently and are all exhausted in the left

  26. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    because each maps differently, is why it is injective The fact that each gets used up is actually a property of it being a function.

  27. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Ok @GIL.ojei is it surjective

  28. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    gil no two x values map to the same y value. agreed?

  29. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  30. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    OK @GIL.ojei is it surjective?

  31. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no

  32. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    why ot?

  33. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    remember that surjective just means that the entire codomain (the circle on the right) gets used up

  34. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    because no single element of X mspd to two elements of Y

  35. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    read my last comment

  36. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    what you just said is true of all functions

  37. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    It is injective because no two x elements get sent to one y value It is surjective because every element in the codomain gets used up

  38. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh ok

  39. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    ok so when it is both surjective and injective we call it a bijection.

  40. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Here is what is important, and here is the notation that we use A function \(f:D\rightarrow C\) is a relation with the following two properties. 1) \(a=b\implies f(a)=f(b)\) 2) \(\forall x\in D\) it is true that \(f(x)\in C\) A function is surjective if \(f(a)=f(b)\implies a=b\) where \(a,b\in D\). A function is surjective if \(\forall y\in C \ \exists \ x\in D\) such that \(f(x) = y\). A function that is both surjective and injective is called a bijection.

  41. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  42. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Now think about why these things are saying what we talked about above.

  43. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    @jayzdd pointed out, in my very first post I should have said injective implies no two domain elements map to a single codomain element

  44. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok buy i do not understand this

  45. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    A function \(f:D\rightarrow C\) is a relation with the following two properties. 1) \(a=b\implies f(a)=f(b)\) 2) \(\forall x\in D\) it is true that \(f(x)\in C\) A function is surjective if \(f(a)=f(b)\implies a=b\) where \(a,b\in D\). A function is surjective if \(\forall y\in C \ \exists \ x\in D\) such that \(f(x) = y\). A function that is both surjective and injective is called a bijection.

  46. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    buy ?

  47. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but

  48. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Ok well lets talk about the surjective surjective says that no two different domain elements can map to the same y value, so if two things map to the same y value they better be the same thing in other words \(f(a) = f(b) \implies a=b\)

  49. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  50. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    does that make sense?

  51. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    The reason we are going over this part is because you are going to be asked to show a function is surjective and injective and this is how you do it.

  52. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes thanks

  53. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    when will you be online again sir?

  54. zzr0ck3r
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    most likely

  55. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.