Answer :
To solve this problem, we need to analyze the elements of the set [tex]\( A \)[/tex] and explore their relationships with different sets of numbers. The elements in set [tex]\( A \)[/tex] are a mix of integers, rational numbers, irrational numbers, and possibly other real numbers.
Here's the set [tex]\( A \)[/tex]:
[tex]\[
A = \{-\sqrt{25}, +\sqrt{64}, -\frac{3}{2}, \frac{48}{12}, -0.8, \frac{\sqrt{81}}{3}, -0.5, -\sqrt{(-2)^6 \cdot (-5)^2}, \sqrt{2^4 \cdot 3^4}, \sqrt{12}, -\sqrt{24}, +\sqrt{48}, -\sqrt{16}\}
\][/tex]
We'll evaluate each element of [tex]\( A \)[/tex]:
- [tex]\(-\sqrt{25} = -5\)[/tex]
- [tex]\(+\sqrt{64} = 8\)[/tex]
- [tex]\(-\frac{3}{2} = -1.5\)[/tex]
- [tex]\(\frac{48}{12} = 4\)[/tex]
- [tex]\(-0.8 = -0.8\)[/tex]
- [tex]\(\frac{\sqrt{81}}{3} = 3\)[/tex]
- [tex]\(-0.5 = -0.5\)[/tex]
- [tex]\(-\sqrt{(-2)^6 \cdot (-5)^2} = -40\)[/tex]
- [tex]\(\sqrt{2^4 \cdot 3^4} = 36\)[/tex]
- [tex]\(\sqrt{12} \approx 3.464\)[/tex]
- [tex]\(-\sqrt{24} \approx -4.899\)[/tex]
- [tex]\(+\sqrt{48} \approx 6.928\)[/tex]
- [tex]\(-\sqrt{16} = -4\)[/tex]
Let's perform the calculations as asked:
1. [tex]\( A \cap \mathbb{N} \)[/tex] (Natural numbers):
[tex]\[
\{3, 4, 8\}
\][/tex]
These are the numbers in [tex]\( A \)[/tex] that are also natural numbers.
2. [tex]\( A \cap \mathbb{Z} \)[/tex] (Integers):
[tex]\[
\{-5, -4, 3, 4, 8, -40, 36\}
\][/tex]
These are the numbers in [tex]\( A \)[/tex] that are integers.
3. [tex]\( A \cap \mathbb{Q} \)[/tex] (Rational numbers):
[tex]\[
\{-5, -4, -1.5, -0.8, -0.5, 3, 4, 8, 36, -40\}
\][/tex]
These are numbers in [tex]\( A \)[/tex] that are rational.
4. [tex]\( A \backslash \mathbb{Q} \)[/tex] (Irrational numbers):
[tex]\[
\{\sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928\}
\][/tex]
These are numbers in [tex]\( A \)[/tex] that are irrational.
5. [tex]\( A \backslash \mathbb{R} \)[/tex] (Empty set since all elements are real):
[tex]\[
\{\}
\][/tex]
6. [tex]\( A \cap (\mathbb{R} \backslash \mathbb{Q}) \)[/tex] (Real but not rational numbers):
[tex]\[
\{\sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928\}
\][/tex]
7. [tex]\( A \backslash \mathbb{Z} \)[/tex] (Non-integer numbers):
[tex]\[
\{-1.5, -0.8, -0.5, \sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928\}
\][/tex]
8. Original set [tex]\( A \)[/tex]:
[tex]\[
\{-5, 8, -1.5, 4, -0.8, 3, -0.5, -40, 36, \sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928, -4\}
\][/tex]
This breakdown illustrates how each element of [tex]\( A \)[/tex] fits into various categories of numbers.
Here's the set [tex]\( A \)[/tex]:
[tex]\[
A = \{-\sqrt{25}, +\sqrt{64}, -\frac{3}{2}, \frac{48}{12}, -0.8, \frac{\sqrt{81}}{3}, -0.5, -\sqrt{(-2)^6 \cdot (-5)^2}, \sqrt{2^4 \cdot 3^4}, \sqrt{12}, -\sqrt{24}, +\sqrt{48}, -\sqrt{16}\}
\][/tex]
We'll evaluate each element of [tex]\( A \)[/tex]:
- [tex]\(-\sqrt{25} = -5\)[/tex]
- [tex]\(+\sqrt{64} = 8\)[/tex]
- [tex]\(-\frac{3}{2} = -1.5\)[/tex]
- [tex]\(\frac{48}{12} = 4\)[/tex]
- [tex]\(-0.8 = -0.8\)[/tex]
- [tex]\(\frac{\sqrt{81}}{3} = 3\)[/tex]
- [tex]\(-0.5 = -0.5\)[/tex]
- [tex]\(-\sqrt{(-2)^6 \cdot (-5)^2} = -40\)[/tex]
- [tex]\(\sqrt{2^4 \cdot 3^4} = 36\)[/tex]
- [tex]\(\sqrt{12} \approx 3.464\)[/tex]
- [tex]\(-\sqrt{24} \approx -4.899\)[/tex]
- [tex]\(+\sqrt{48} \approx 6.928\)[/tex]
- [tex]\(-\sqrt{16} = -4\)[/tex]
Let's perform the calculations as asked:
1. [tex]\( A \cap \mathbb{N} \)[/tex] (Natural numbers):
[tex]\[
\{3, 4, 8\}
\][/tex]
These are the numbers in [tex]\( A \)[/tex] that are also natural numbers.
2. [tex]\( A \cap \mathbb{Z} \)[/tex] (Integers):
[tex]\[
\{-5, -4, 3, 4, 8, -40, 36\}
\][/tex]
These are the numbers in [tex]\( A \)[/tex] that are integers.
3. [tex]\( A \cap \mathbb{Q} \)[/tex] (Rational numbers):
[tex]\[
\{-5, -4, -1.5, -0.8, -0.5, 3, 4, 8, 36, -40\}
\][/tex]
These are numbers in [tex]\( A \)[/tex] that are rational.
4. [tex]\( A \backslash \mathbb{Q} \)[/tex] (Irrational numbers):
[tex]\[
\{\sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928\}
\][/tex]
These are numbers in [tex]\( A \)[/tex] that are irrational.
5. [tex]\( A \backslash \mathbb{R} \)[/tex] (Empty set since all elements are real):
[tex]\[
\{\}
\][/tex]
6. [tex]\( A \cap (\mathbb{R} \backslash \mathbb{Q}) \)[/tex] (Real but not rational numbers):
[tex]\[
\{\sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928\}
\][/tex]
7. [tex]\( A \backslash \mathbb{Z} \)[/tex] (Non-integer numbers):
[tex]\[
\{-1.5, -0.8, -0.5, \sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928\}
\][/tex]
8. Original set [tex]\( A \)[/tex]:
[tex]\[
\{-5, 8, -1.5, 4, -0.8, 3, -0.5, -40, 36, \sqrt{12} \approx 3.464, -\sqrt{24} \approx -4.899, \sqrt{48} \approx 6.928, -4\}
\][/tex]
This breakdown illustrates how each element of [tex]\( A \)[/tex] fits into various categories of numbers.
