Let $f:A \to B$. Show that if $X \subseteq A$, then $X \subseteq f^{-1}(f(X))$.
158982
Image, pre-image, and injection proof verification
138105
Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
97184
Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
79281
Let $f : X \to Y$ be a function and $E \subseteq X$ and $F \subseteq X$. Show that in general
47135
Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
18905
$f(a)=f(b)=f_{+}'(a)=f_{-}'(b)=0$ and $|f''(x)|\le M$. Show that $|f(x)|\le \frac{M(b-a)^2}{16}$.
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
7858
Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$
64905
Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
46056
If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
121771
Proof $f(x)\equiv 0$
41477
Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
49143
Showing $A \subset B \iff A\cap B=A$
139111
Prove that $a<c<b$, $\frac{f(b)-f(a)-(b-a)f'(a)}{g(b)-g(a)-(b-a)g'(a)}=\frac{f"(c)}{g"(c)}$
75863
how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$
80421
Show that $a - b \mid f(a) - f(b)$
116056
Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.