1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Injective/Surjective Functions

  1. Sep 18, 2012 #1

    dpa

    User Avatar

    1. The problem statement, all variables and given/known data
    Is the minimum function defined by f(a,b)=min{a,b} surjective or injective?


    2. Relevant equations
    a function is injective f(x)=f(y) always implies x=y.
    a function is surjective if for every y in codomain, there exists an x in domain such that f(x)=y.


    3. The attempt at a solution
    I am confused whether min is surjective function or not.
    As for injective, it is not. e.g. f(2,3) and f(3,2) both give 2. This is sufficient to say it is not injective.
    But it is surjective, which I am mostly sure, but how do I show it is surjective?
     
  2. jcsd
  3. Sep 18, 2012 #2

    dpa

    User Avatar

    And, is the following the right way to show that the function is surjective?
    f(x)=y,
    x=f^-1(y)
    f(f^-1(y))=x
    Is this why it is called right invertible?
     
  4. Sep 18, 2012 #3

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You're correct that it is not injective. Whether or not it is surjective depends on the codomain. Since the codomain is not specified here, there's no way to answer the question. Any function is surjective if you define its codomain to be its image.
     
  5. Sep 18, 2012 #4

    dpa

    User Avatar

    Sorry, that I forgot the first part.
    It is defined for all f:ZXZ gives z. So c0 domain is set of all integers.
    I am supposed to prove it not just explain.

    Thank You.
     
  6. Sep 18, 2012 #5

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    OK, that makes it easy to answer whether the function is surjective. Given an arbitrary integer n, can you find two integers, a and b, such that min(a, b) = n?
     
  7. Sep 18, 2012 #6

    dpa

    User Avatar

    Yes, definitely I mean for any integer, thats possible unless n=infinity which I believe does not belong to Z.
    So, does verbal proof suffice?

    Thank You.
    :-)
     
  8. Sep 18, 2012 #7

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Right, Z is the set of all integers. Infinity is not an integer.

    Why don't you write down your proposed verbal proof and we'll see how it looks. Ideally (for the surjective part), can you name specific values for a and b such that min(a,b) = n?
     
  9. Sep 18, 2012 #8

    dpa

    User Avatar

    Verbal Part:
    We know that for any value of an integer, we can find two integers such that the smallest of those two integers is the first integer. i.e. for every integer n we can write integers n and n+a where, a>=0. which gives min{n,n+a}=n.
    Specific example would be for n=100, we can write two integers 100 and 101. I doubt if it is ideal example.
     
  10. Sep 18, 2012 #9

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes, that works. Note that there's nothing requiring the two integers to be different from each other. You also have min(n, n) = n.
     
  11. Sep 18, 2012 #10

    dpa

    User Avatar

    Thank You.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Injective/Surjective Functions
Loading...