Show that $S$ is subset of f$^{-1}(f(S))$
127285
Inverse images of images, images of inverse images of a set of vectors
132541
Showing ${1\over n}\sum|S_i|=O(\sqrt n)$ for $S_i\subset [n]$, $|S_i\cap S_i|\le 1$ for $i\ne j$
23125
Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.
17461
Show that a set is dense in $[-1,1]$
41477
Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
106487
Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$
65923
How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
101778
Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
40604
Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?
108633
Why $f^{-1}(f(A)) \not= A$
19856
Prove that a subset of a separable set is itself separable
49592
Show that, if $f:A\to B$ is a function, with $A$ and $B$ being finite sets, and $|A|=|B|$, then $f$ is one to one iff $f$ is onto.
95575
If $f(s) = (1+s)^{(1+s)^{(1+s)/s}/s}$, show that $\lim_{s \to \infty} f(s)/s = 1$.
101371
show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$
151719
Connected set in $\mathbb{S}^1$
111538
If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?
27482
How to show $-\sup(-A)=\inf(A)$?