query: $\phi:A\to B$, $F:C^B \to C^A$. $F(f)=f\circ\phi$. prove that if $\phi$ is injective, then F is surjective
positive_passages.docid: 28798
positive_passages.text: Proving injection and surjection with functions $F:C^B\to C^A$, $F(f)=f\circ\varphi$, $\varphi: A\to B$
negative_passages.docid: 123467
negative_passages.text: Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$.
negative_passages.docid: 89514
negative_passages.text: Show that if $g \circ f$ is injective, then so is $f$.
negative_passages.docid: 10253
negative_passages.text: Proving that all entire & injective functions take the form $f = ax + b$?
negative_passages.docid: 76610
negative_passages.text: Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
negative_passages.docid: 40868
negative_passages.text: Composition of two functions - Bijection
negative_passages.docid: 76674
negative_passages.text: Let $S:U\rightarrow V, \ T:V\rightarrow W$ and if $S$ and $T$ are both injective/surjective, is $T\circ S$ injective/surjective?
negative_passages.docid: 82448
negative_passages.text: Why is $f(x) = x\phi(x)$ one-to-one?
negative_passages.docid: 93040
negative_passages.text: $f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki
negative_passages.docid: 63146
negative_passages.text: Proof of properties of injective and surjective functions.
negative_passages.docid: 95299
negative_passages.text: What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15
negative_passages.docid: 2072
negative_passages.text: Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
negative_passages.docid: 22061
negative_passages.text: $|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.
negative_passages.docid: 40383
negative_passages.text: Is it true that $|g| = |\phi(g)|$ for all homomorphisms $\phi: G \to G$ and $g \in G$?
negative_passages.docid: 127488
negative_passages.text: If $L:\mathbb{R}^n \rightarrow \mathbb{R}^m$ and $m<n$ linear trans. What does this tell about injectivity and surjectivity?
negative_passages.docid: 24107
negative_passages.text: Bijection $f:\mathcal{P}(A)\to(A\to \{0,1\})$
negative_passages.docid: 88201
negative_passages.text: Suppose $X$ is a finite set and $f : X \to X$ is a function. Then $f$ is injective if and only if $f$ is surjective.