Given set $A = \{-\sqrt{10}, -3, -\frac{5}{2}, -0.025, 0, \sqrt{2}, -\frac{8}{2}, e^2 \}$, let's identify which elements belong to each of the following sets:

a. The Set of Real Numbers

The set of real numbers includes all numbers that are not imaginary or complex. Since all the elements of $A$ are real, we have: $\{-\sqrt{10}, -3, -\frac{5}{2}, -0.025, 0, \sqrt{2}, -\frac{8}{2}, e^2 \}$

b. The Set of Natural Numbers

Natural numbers are positive integers (starting from 1). There are no natural numbers in $A$ because all elements are either negative, zero, or non-integers.

c. The Set of Whole Numbers

Whole numbers include all non-negative integers (0, 1, 2, 3, ...). The only whole number in set $A$ is: $\{0\}$

d. The Set of Integers

Integers include both positive and negative whole numbers. The elements of $A$ that are integers are: $\{-3, 0, -\frac{8}{2} = -4\}$

e. The Set of Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. The irrational elements of $A$ are: $\{-\sqrt{10}, \sqrt{2}, e^2\}$

Summary of Elements in Each Set:

• a. Real Numbers: $\{-\sqrt{10}, -3, -\frac{5}{2}, -0.025, 0, \sqrt{2}, -\frac{8}{2}, e^2 \}$
• b. Natural Numbers: $\{\}$ (none)
• c. Whole Numbers: $\{0\}$
• d. Integers: $\{-3, 0, -4\}$
• e. Irrational Numbers: $\{-\sqrt{10}, \sqrt{2}, e^2\}$

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Would you like more details or have any questions?

Here are five related questions:

1. What is the difference between rational and irrational numbers?
2. Can an irrational number be expressed as a repeating decimal?
3. How do you identify if a number is an integer?
4. What makes a number a whole number, and how do they relate to natural numbers?
5. Why is $e^2$ considered an irrational number?

Tip: An irrational number has a non-repeating, non-terminating decimal expansion.