With only one of; Singleton, Empty or Infinite, classify the following sets and give
a reason why;
(i) X = {x : −2 < x < 0 and x ∈ N} (1)
(ii) Y = {x : x is an even number, and x is also prime} (1)
(iii) Z = {1, 2, 3, . . .} (1)
(iv) The set of all straight lines parallel to the y−axis on the x, y−plane.

To classify each set as either *Singleton*, *Empty*, or *Infinite*, let's analyze them one by one:

### (i) \( X = \{x : -2 < x < 0 \text{ and } x \in \mathbb{N} \} \)
- **Explanation:** The set asks for natural numbers (\(\mathbb{N}\)) that lie between -2 and 0. However, the natural numbers start at 1 and do not include negative numbers or 0. Therefore, no natural number satisfies \(-2 < x < 0\).
- **Classification:** **Empty set** (since there are no elements in this set).

### (ii) \( Y = \{x : x \text{ is an even number, and } x \text{ is also prime} \} \)
- **Explanation:** Prime numbers are numbers greater than 1 with no divisors other than 1 and themselves. The only even prime number is 2, because all other even numbers are divisible by 2 and thus not prime.
- **Classification:** **Singleton set** (since the only element of this set is 2).

### (iii) \( Z = \{1, 2, 3, \dots\} \)
- **Explanation:** This set is the set of all natural numbers, which continues indefinitely. It has no upper bound.
- **Classification:** **Infinite set** (since it contains an unbounded, endless number of elements).

### (iv) The set of all straight lines parallel to the y-axis on the \(x, y\)-plane.
- **Explanation:** Any line parallel to the y-axis can be described by the equation \(x = c\), where \(c\) is a constant. Since \(c\) can be any real number, there are infinitely many such lines.
- **Classification:** **Infinite set** (since there are infinitely many lines parallel to the y-axis).

### Summary of Classifications:
1. \(X = \emptyset\) → **Empty set**
2. \(Y = \{2\}\) → **Singleton set**
3. \(Z = \{1, 2, 3, \dots\}\) → **Infinite set**
4. Set of straight lines parallel to the y-axis → **Infinite set**

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Would you like any details on these classifications?

Here are five related questions to explore further:
1. What is the definition of a natural number?
2. Can a set be both finite and infinite? Why or why not?
3. What properties define a prime number?
4. How are parallel lines defined mathematically in the Cartesian plane?
5. What is the difference between an empty set and a singleton set?

**Tip:** A set is classified as empty when it contains no elements, and as singleton if it contains exactly one element!

With only one of; Singleton, Empty or Infinite, classify the following sets and give a reason why; (i) X = {x : −2 < x < 0 and x ∈ N} (1) (ii) Y = {x : x is an even number, and x is also prime} (1) (iii) Z = {1, 2, 3, . . .} (1) (iv) The set of all straight lines parallel to the y−axis on the x, y−plane.

Learn how to classify sets as Singleton, Empty, or Infinite by analyzing specific examples. This problem explores natural numbers, prime numbers, and geometry with parallel lines on the Cartesian plane. Suitable for students in Grades 9-12.

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