Results 1 to 2 of 2

Math Help - Injectivity and Surjectivity

  1. #1
    Member
    Joined
    Aug 2007
    Posts
    239

    Injectivity and Surjectivity

    Let  f: X \to Y be a function. Prove that (i)  f is injective  \Leftrightarrow \overrightarrow{f} is injective  \Leftrightarrow \overleftarrow{f} is surjective.

    (ii)  f is surjective  \Leftrightarrow  \overleftarrow{f} is surjective  \Leftrightarrow \overleftarrow{f} is injective.

    So for (i)  \forall x_1, x_2 \in X, (f(x_1) = f(x_2) \Rightarrow x_1 = x_2 ). So  \overrightarrow{f}(A) = \{f(x)| x \in A \} for  A \in \mathcal{P}(X) . Say  x_{1} \in A and  x_2 \in A . Then  f(x_1) = f(x_2) \Rightarrow x_1 = x_2 .  \overleftarrow{f}(B) = \{x \in X| f(x) \in B \} for  B \in \mathcal{P}(Y) and so  \forall y \in Y,  \exists x \in X, f(x) = y which implies surjectivity.

    So then you go in the opposite direction to complete the proof?

    Is it a similar case for (ii)?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2007
    Posts
    239
    Did I get the general proof correct?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. weaker projectivity/injectivity
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 12th 2012, 02:14 AM
  2. Set Theory: Functions (injectivity/surjectivity)
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 10th 2010, 03:33 PM
  3. composite functions and their injectivity and surjectivity
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: January 8th 2010, 10:54 AM
  4. Theorem on Surjectivity and Injectivity
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: February 25th 2009, 09:56 PM
  5. injectivity and surjectivity (onto and one-one)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 22nd 2009, 09:01 PM

Search Tags


/mathhelpforum @mathhelpforum