Can someone show me how to prove that $f:X \rightarrow Y$ is onto if and only if it has a right inverse?
78698
Prove if $f$ has a right inverse function: $f\circ g=id_x$ $\iff$ $f$ is onto $Y$
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.
10911
$A \in M_{n}(\mathbb{R})$ has inverse if and only if it's rank is n.
37308
Let $f:K\to K$ with $\|f(x)-f(y)\|\geq ||x-y||$ for all $x,y$. Show that equality holds and that $f$ is surjective.
48990
Linear map $f:V\rightarrow V$ injective $\Longleftrightarrow$ surjective
92376
Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
76674
Let $S:U\rightarrow V, \ T:V\rightarrow W$ and if $S$ and $T$ are both injective/surjective, is $T\circ S$ injective/surjective?
21891
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
104757
If $f \circ g$ is surjective, then f is surjective
101773
Let $g : A \to B$ and $f : B \to C$ be functions. Let $f \circ g$ be onto. Are both $f$ and $g$ necessarily onto?
93712
Let $f:(-1,1) \rightarrow \mathbb{R}$ be smooth on $(-1,1)$, $f(0) = 1$. Find $f(x)$.
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 $f^{-1}$.
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
139537
Show that $x\subseteq y \Leftrightarrow \mathcal{P}(x)\subseteq\mathcal{P}(y)$.
10024
How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition)
152841
I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
3820
Prove that $||x|-|y||\le |x-y|$