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Thread: Injectivity and Surjectivity

  1. #1
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    Injectivity and Surjectivity

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

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

    So for (i) $\displaystyle \forall x_1, x_2 \in X, (f(x_1) = f(x_2) \Rightarrow x_1 = x_2 )$. So $\displaystyle \overrightarrow{f}(A) = \{f(x)| x \in A \} $ for $\displaystyle A \in \mathcal{P}(X) $. Say $\displaystyle x_{1} \in A $ and $\displaystyle x_2 \in A $. Then $\displaystyle f(x_1) = f(x_2) \Rightarrow x_1 = x_2 $. $\displaystyle \overleftarrow{f}(B) = \{x \in X| f(x) \in B \} $ for $\displaystyle B \in \mathcal{P}(Y) $ and so $\displaystyle \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)?
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  2. #2
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    Did I get the general proof correct?
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