1. Stating the problem: We analyze multiple points, sets, operations, and relations involving coordinates, sets, intersections, and arithmetic given in the user's input. 2. Points given in coordinate plane: A$(\sqrt{2},201)$, B$(-\sqrt{2},|{-20}|)$ which simplifies to B$(-\sqrt{2},20)$ since absolute value makes $|-20|=20$. 3. Coordinates given as $(-\frac{1}{2},-\frac{1}{2})$, $(-\frac{1}{2}, \frac{1}{2})$, $(\frac{1}{2}, \frac{1}{2})$ likely represent vertices in unit lengths. 4. Set $E = \{-\sqrt{3}, \frac{22}{7}, \sqrt{\frac{12}{27}}, \frac{\pi}{3}, \sqrt{0.90}, 0.12131415\}$. - Simplify $\sqrt{\frac{12}{27}} = \sqrt{\frac{4}{9}} = \frac{2}{3}$. - Simplify $\sqrt{0.90} = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} \approx 0.9487$. 5. Intersecting $E$ with intervals $I$, integers $\mathbb{Z}$, rationals $\mathbb{Q}$: - $E \cap I$: $I$ is interval; here, possibly real numbers including those values; all elements of $E$ are numbers; so $E \cap I = E$ (since all are real numbers). - $E \cap \mathbb{Z}$ (integers): From $E$, only integer candidate is $-\sqrt{3}$ (irrational), $22/7$ (rational but not integer), $2/3$ (rational no), $\frac{\pi}{3}$ irrational, $\sqrt{0.90}$ irrational, $0.12131415$ decimal – none are integers; so $E \cap \mathbb{Z} = \emptyset$. - $E \cap \mathbb{Q}$ (rationals): Rational numbers here include $\frac{22}{7}$ (approximate pi, rational), $\frac{2}{3}$ (simplified), also $0.12131415$ is decimal, is rational (terminating decimal), so yes. $-\sqrt{3}$ and $\frac{\pi}{3}$ are irrational, $\sqrt{0.90}$ irrational. Thus $E \cap \mathbb{Q} = \{\frac{22}{7}, \frac{2}{3}, 0.12131415\}$. 6. Given $x = 3.12345$, then $10x = 10 \times 3.12345 = 31.2345$. 7. Points: - $N(3,-2)$, $A(-3,0)$, $M(3,2)$. - Axes OI and OJ both length 1 cm and perpendicular. 8. Showing $(MN) \parallel (OJ)$: - Find vector $\overrightarrow{MN} = (3 - 3, 2 - (-2)) = (0,4)$. - $OJ$ is axis along y (vertical) of length 1 cm. - Vector $OJ = (0,1)$. - Since both vectors are vertical and scalar multiples, $\overrightarrow{MN} \parallel \overrightarrow{OJ}$. Final answers: - $B$ is $(-\sqrt{2},20)$ - $E = \{-\sqrt{3}, \frac{22}{7}, \frac{2}{3}, \frac{\pi}{3}, \sqrt{0.90}, 0.12131415\}$ - $E \cap I = E$ - $E \cap \mathbb{Z} = \emptyset$ - $E \cap \mathbb{Q} = \{\frac{22}{7}, \frac{2}{3}, 0.12131415\}$ - $10x = 31.2345$ - $(MN) \parallel (OJ)$ is true because both are vertical vectors.