Homework 10

1. Prove or disprove that $\{m:\mathbb{Z} \mid m\ge 10\}\subseteq \{n:\mathbb{Z} \mid n^3-6n^2\geq 4n\}$.

2. Prove or disprove that $\{t:\mathbb{R} \mid t^2-3t+2=0\}=\{s:\mathbb{R} \mid s=2\}$.

3. Prove or disprove that $\{1,2,3\}\cap\{2,3,4\}=\{2,3,6\}$.

   Write this problem only in Lean, not on paper. Don't use the tactic exhaust (which can fill in most of the solution automatically) -- do it all by hand.

4. Prove that $\{r:\mathbb{Z}\mid r\equiv 11\mod 15\}=\{s:\mathbb{Z}\mid s\equiv 2\mod 3\}\cap\{t:\mathbb{Z}\mid t\equiv 1\mod 5\}$.

5. Determine which of these properties hold for the relation $\sim$ on $\mathbb{Z}$, defined by, $a\sim b$ if there exist positive integers $m$ and $n$ such that $am=bn$:

   a. reflexive

   b. symmetric

   c. antisymmetric

   d. transitive

   Also (for submission only on paper, not in Lean)

   e. Sketch (a representative portion of) this relation as a directed graph.

6. Determine which of these properties hold for the relation $\prec$ on $\mathbb{R}^2$, defined by, $(x_1,y_1)\prec (x_2,y_2)$ if $x_1\le x_2$ and $y_1 \le y_2$:

   a. reflexive

   b. symmetric

   c. antisymmetric

   d. transitive