$|A|=|B|= n \in \omega $ and $f: A \rightarrow B$ an injective (surjective) function then $f$ is surjective (injective) function.
122814
Finite Sets, Equal Cardinality, Injective $\iff$ Surjective.
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.
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?
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
45792
Rigorous proof that surjectivity implies injectivity for finite sets
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
143618
If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)
14975
Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$
91852
A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.
16413
Proving existence of surjective $f:\mathbb{N} \rightarrow A$ implies $A$ is at most countable.
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
20303
A function $f\colon E\to F$ between finite sets is injective iff it is surjective iff it is bijective
98570
Prove whether $f(n)$ is $O$, $o$, $\Omega$, $\omega$ or $\Theta$ of $g(n)$. $f(n) = n + (\log n)^{2}$, $g(n) = n + \log(n^{2})$.
106830
are $f(f(x))=f(x)$ and $f(x)=x$ equivalent?
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.
136056
$ f:\mathbb C\rightarrow \mathbb C$ defined by $f(z)=0$, if $Re(z)=0$ or $Im(z)=0$ and $f(z)=z$, otherwise.
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?