Show that $f(A \cup B) \subseteq f(A) \cup f(B)$, $f(A) \setminus f(B) \subseteq f(A \setminus B)$
1821
Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality
102168
Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
148668
How to show $A\times (B\cup C) = (A\times B)\cup (A\times C)$
49143
Showing $A \subset B \iff A\cap B=A$
28641
Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected
28391
Prove the compactness of a set B
80421
Show that $a - b \mid f(a) - f(b)$
83714
$A \in B$ vs. $A \subset B$ for proofs
125043
Show that F(A−B) is not necessarily a subset of F(A)−F(B).
33524
Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
75966
Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements.
30694
Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
85465
Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
155999
How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$
144693
Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$
93157
If $A \subseteq B$, does $\mbox{dist}(x,\partial A) \le \mbox{dist}(x, \partial B)$ hold for all $x \in A$