for functions $f: A \to B$,$g:P(A) \to P(B)$ when $g(Y) = f^{-1}[Y]$, prove that $f$ is surjective iff $g$ is injective
121395
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
88201
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.
136842
$|x-f(x)|<5777$, Show that $f$ is surjective
47135
Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
53101
Is it true that the relation |A| < |B| is a sufficient condition for claiming that $f$ is a bijection?
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
40868
Composition of two functions - Bijection
157756
Examine if function $f:\mathbb{R^2}\rightarrow \mathbb{R^2}$ which is defined as $f(x,y)=(2x-y,x-4y)$ is bijective. If bijective, find $f^{-1}$.
56413
$f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
116732
Proving that a continuous map is surjective
107659
Let $f:A \to B$ and $g:B\to A$ be arbitrary functions.
21707
Constructing a Bijective function via an injective and surjective one.
46863
Show that $g$ is injective
146188
Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice?
21391
Proving Functions are Surjective
53168
Surjective, Injective, Bijective Functions from $\mathbb Z$ to itself
49882
Proving $f(x,y,z)=\big(\frac{x}{a+x+y+z}, \frac{y}{a+x+y+z}, \frac{z}{a+x+y+z} \big)$ is injective