1. Stating the problem: We analyze the set $E = \{-\sqrt{3}; \frac{22}{7}; \sqrt{\frac{12}{27}}; \frac{\pi}{3}; \sqrt{0.90}; 0.12131415\}$ and points $A(3,2), M(3,2), N(-3,0)$, with segment $MN$ parallel to line $OJ$ where $OI$ and $OJ$ are perpendicular, each 1 cm. 2. Simplify elements of set $E$: - $-\sqrt{3}$ remains $-\sqrt{3}$. - $\frac{22}{7}$ is an approximation of $\pi$. - $\sqrt{\frac{12}{27}}$ simplifies: $\frac{12}{27} = \frac{4}{9}$, so $\sqrt{\frac{12}{27}} = \sqrt{\frac{4}{9}} = \frac{2}{3}$. - $\frac{\pi}{3}$ is approximately 1.047. - $\sqrt{0.90} \approx 0.9487$. - $0.12131415$ is a decimal number. 3. Find intersections of $E$ with known sets: - $E \cap \mathbb{I}$ (irrational numbers): includes $-\sqrt{3}$ and $\pi/3$. - $E \cap \mathbb{Z}$ (integers): none, no integers in $E$. - $E \cap \mathbb{Q}$ (rationals): $\frac{22}{7}$ and $\frac{2}{3} = \sqrt{12/27}$ are rational. 4. Analyze points $A(3,2)$, $M(3,2)$, $N(-3,0)$ and segment $MN$ parallel to line $OJ$: - Vector $\overrightarrow{MN} = (3 - (-3), 2 - 0) = (6,2)$. - Since $OI$ and $OJ$ are perpendicular unit segments along axes, assume $OJ$ along x-axis: direction vector is $(1,0)$. - $\overrightarrow{MN} = (6,2)$ is not parallel to $OJ$ as $(6,2)$ is not scalar multiple of $(1,0)$. - If $OJ$ along y-axis $(0,1)$, again no parallelism since $(6,2)$ not scalar multiple of $(0,1)$. 5. Possibly $OJ$ direction vector differs; with data given, segment $MN$ is not parallel to unit $OJ$ along axes. Final results: - Simplified $E = \{-\sqrt{3}, \frac{22}{7}, \frac{2}{3}, \frac{\pi}{3}, \sqrt{0.90}, 0.12131415\}$. - Irrationals in $E$: $\{-\sqrt{3}, \frac{\pi}{3}\}$. - Rationals in $E$: $\{\frac{22}{7}, \frac{2}{3}, 0.12131415\}$ (assuming decimal is rational). - No integers in $E$. - $\overrightarrow{MN} = (6,2)$ is not parallel to $OJ$ if $OJ$ along standard axes.