Definition 3.3.17 (Onto functions). A function \(f\) is onto (or surjective) if \(f(X)=Y\), i.e., every element in \(Y\) comes from applying \(f\) to some element in \(X\): $$ \text{For every } y\in Y,\text{ there exists } x\in X \text{ such that } f(x)=y. $$
Definition 3.3.17
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Exercise 3.4.1
\(=\) { \(f\) is bijective \(\Rightarrow f\) is surjective
\(\quad~~\) Definition 3.3.17: \(\forall y\in V, \exists x\in X, f(x)=y\) }\(=\) { \(f\) is bijective \(\Rightarrow f^{-1}\) is surjective
\(\quad~~\) Definition 3.3.17: \(\forall f^{-1}(y)\in X, \exists y'\in Y, f^{-1}(y')=f^{-1}(y)\) }\(=\) { \(f\) is bijective \(\Rightarrow f^{-1}\) is surjective
\(\quad~~\) Definition 3.3.17: \(\forall x\in X, \exists y\in Y, f^{-1}(y)=x\) } -
Exercise 3.3.7
\(\equiv\) { Definition 3.3.17 }
- Exercise 3.3.5
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Exercise 3.3.3
\(\equiv\) { Definition 3.3.17 }
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Exercise 3.3.2
\(\vdash\) { Definition 3.3.17 }