If $A \subseteq B, $ then $f^{-1}(A) \subseteq f^{-1}(B)$
17212
The preimage of a subset
80009
Prove that $| |a|-|b| | \le |a-b|$
25710
$A ⊂ B$ if and only if $A − B = ∅$
113411
If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$?
123275
Prove $[A,B^n] = nB^{n-1}[A,B]$
62172
Why does $-b < a < b \iff |a| < b$?
9854
If $U$ is a neighborhood of $e$, there exists $V$ neighborhood of $e$ such that $V^{-1}V \subseteq U$
145125
How to prove that $A^{-1} + B^{-1}$ is invertible given the conditions
47856
$f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
50655
Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$
80421
Show that $a - b \mid f(a) - f(b)$
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.
68781
proving if $a \le b$ then $\sup(A) \le \inf(B)$
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
80827
Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
40604
Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?
1821
Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality