Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    0
    ReputationRep:
    If there is a function f: X -> Y where X is countable, then there is a bijection g: N -> X. What I don't understand is why is g(f(1)), g(f(2)),... a list of elements of Y? Can someone explain this to me please.
    EDIT: f:X -> Y is a surjective map.
    Offline

    16
    ReputationRep:
    I don't understand that either. Firstly, f(1) isn't defined, since 1 isn't necessarily in X. Secondly, because we need some more conditions on f (even if we were taking f(g(1)), f(g(2)), ...
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by SimonM)
    I don't understand that either. Firstly, f(1) isn't defined, since 1 isn't necessarily in X. Secondly, because we need some more conditions on f (even if we were taking f(g(1)), f(g(2)), ...
    Oh yeah! I forgot to mention that f: X -> Y is surjective.
    • Thread Starter
    Offline

    0
    ReputationRep:
    No one?
    Offline

    2
    ReputationRep:
    If X is countable then X ~ N. If f:N->X is subjective then X is countable.

    ..

    Spoiler:
    Show
    the proof of the second statement is demi non trivial as a tip you would be wise to consider the set {n | f(n) = x} where x is an element of X.
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by DeanK22)
    If X is countable then X ~ N. If f:N->X is subjective then X is countable.

    ..

    Spoiler:
    Show
    the proof of the second statement is demi non trivial as a tip you would be wise to consider the set {n | f(n) = x} where x is an element of X.
    Sorry I'm not sure what you mean, I just needed to know why g(f(1)),... is a list of elements of Y. Unless of course that was the explanation but I don't understand it.
    Offline

    2
    ReputationRep:
    Why does 1 happen to be in X?
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by DeanK22)
    Why does 1 happen to be in X?
    I'm not sure, but this is the answer "If X is finite then so is Y because f is surjective, hence countable. So we may
    assume X and Y to be infinite. Let g : N -> X be a bijection. Since g is surjective
    g(f(1)), g(f(2)), g(f(3)),... is a list of the elements of Y . Obtain a new list by striking
    out all repetitions in that list and for all n in N, let g(n) be the n-th element in the new list.
    Then g : N -> Y is a bijection, and so Y is countable."
    It doesn't say why 1 is in X and we have to show that Y is countable. I understand the rest of it, just not the part in bold.
    Offline

    2
    ReputationRep:
    Write down everything you know on a piece of paper and think what makes you get to each step. Also, you continually add more and more assumptions! Since when was g bijective? You only mentioned surjectivity.

    Think hard about why 1 does not have to be in X and even if it isn't, why we cannot assume it is (well, we actually won't do any harm in this question but just think ... it is not the case 2 or 3 should be X either ...).
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by DeanK22)
    Write down everything you know on a piece of paper and think what makes you get to each step. Also, you continually add more and more assumptions! Since when was g bijective? You only mentioned surjectivity.

    Think hard about why 1 does not have to be in X and even if it isn't, why we cannot assume it is (well, we actually won't do any harm in this question but just think ... it is not the case 2 or 3 should be X either ...).
    I mentioned that f: X -> Y is a surjective map but the fact that X is countable means there is a function say g for example that g: N -> X is a bijection. Isn't that correct?
    Offline

    2
    ReputationRep:
    I suspect there's a been a typo. f(g(1)), f(g(2)), etc. makes much more sense.
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by Zhen Lin)
    I suspect there's a been a typo. f(g(1)), f(g(2)), etc. makes much more sense.
    Thank you! It's been bothering me for a while now, it seemed very unlikely that it's a typo since it's written in the mark-scheme. But since you said it is a typo I'll believe you.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    What newspaper do you read/prefer?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.