Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$.
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
76610
Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
78698
Prove if $f$ has a right inverse function: $f\circ g=id_x$ $\iff$ $f$ is onto $Y$
28798
Proving injection and surjection with functions $F:C^B\to C^A$, $F(f)=f\circ\varphi$, $\varphi: A\to B$
40604
Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?
22995
Let f:G to H be 1-1 homomorphism show that order of f(a) divides the order of G.
102457
for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
3820
Prove that $||x|-|y||\le |x-y|$
22303
How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
155063
Is Cl(f(A)) a subset of f(Cl(A))?
65923
How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
47135
Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
105455
Prove that "injective function $f:X\to Y$ exists" and "surjective function $g:Y\to X$ exists" is logically equivalent.
152841
I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.