1. The problem involves points A, B, D, and E with given coordinates and values such as AB = 9.6 and AB = 1.6, and some numeric values like 3884, 705.739425, 816.37422, 35.9871, 600, 12. 2. The set E is defined as $E = \{\sqrt{5}, 12.23\}$ with expressions involving square roots, powers of 2, fractions, and decimals. 3. Points are given as $A(2,3)$, $B(2,-1)$, $D(0,3)$, and $E(-\sqrt{2},3)$. 4. We analyze intersections $E \cap \mathbb{Q}$ (rationals), $E \cap \mathbb{Z}$ (integers), and $E \cap D$ (another set, possibly decimals). 5. Expressions for points A and B are: $$A = -(x - 1) + (\pi - 2) - [(\pi - 3) - (\sqrt{2} - 5)]$$ $$B = y - |\sqrt{2} - 5|$$ 6. Simplify expression for A: $$A = -(x - 1) + (\pi - 2) - (\pi - 3) + (\sqrt{2} - 5)$$ $$= -x + 1 + \pi - 2 - \pi + 3 + \sqrt{2} - 5$$ $$= -x + (1 - 2 + 3 - 5) + \sqrt{2}$$ $$= -x - 3 + \sqrt{2}$$ 7. Expression for B: $$B = y - |\sqrt{2} - 5|$$ Calculate $|\sqrt{2} - 5|$: $$\sqrt{2} \approx 1.414$$ $$|1.414 - 5| = | -3.586| = 3.586$$ So, $$B = y - 3.586$$ 8. The problem states $x - y$ and that A and B are opposite (متقابلان), meaning possibly $A = -B$ or $A + B = 0$. 9. Check if $A + B = 0$: $$-x - 3 + \sqrt{2} + y - 3.586 = 0$$ $$-x + y + (\sqrt{2} - 6.586) = 0$$ 10. Rearranged: $$y - x = 6.586 - \sqrt{2}$$ Approximate $\sqrt{2} \approx 1.414$: $$y - x = 6.586 - 1.414 = 5.172$$ 11. Therefore, the relation between $x$ and $y$ is: $$y - x = 5.172$$ or equivalently $$y = x + 5.172$$ 12. For the set intersections: - $E = \{\sqrt{5}, 12.23\}$ - $\sqrt{5}$ is irrational, so $E \cap \mathbb{Q} = \emptyset$ (no rationals) - $12.23$ is decimal, not integer, so $E \cap \mathbb{Z} = \emptyset$ - $E \cap D$ (decimals) includes $12.23$ 13. Summary: - Points and distances given. - Simplified expressions for A and B. - Found linear relation $y = x + 5.172$. - Set intersections analyzed. Final answer: $$y = x + 5.172$$ and $$E \cap \mathbb{Q} = \emptyset, \quad E \cap \mathbb{Z} = \emptyset, \quad E \cap D = \{12.23\}$$