Describe the following sets in roster form?

1. {x/x is a letter of word "Mathematics"}

2. {x/x = 2n, n ∈ N}

3. {x/x3 - 6x² + 11x – 6 = 0}

4. {x/x =1/n, n ∈ N}

Let's convert each of the given sets into **roster form**. Roster form lists all the elements of a set explicitly, separated by commas, and enclosed in curly braces `{}`.

### 1. \(\{x \mid x \text{ is a letter of the word "Mathematics"}\}\)

To list the set in roster form, we need to identify all the unique letters in the word "Mathematics":

- The letters in "Mathematics" are: M, a, t, h, e, m, a, t, i, c, s.
- The unique letters (removing duplicates) are: M, a, t, h, e, i, c, s.

**Roster form**: \(\{M, a, t, h, e, i, c, s\}\)

### 2. \(\{x \mid x = 2n, n \in \mathbb{N}\}\)

This set includes all numbers of the form \(2n\), where \(n\) is a natural number (\(\mathbb{N} = \{1, 2, 3, \ldots\}\)):

- When \(n = 1\), \(x = 2 \times 1 = 2\)
- When \(n = 2\), \(x = 2 \times 2 = 4\)
- When \(n = 3\), \(x = 2 \times 3 = 6\), and so on.

This represents the set of all even natural numbers.

**Roster form**: \(\{2, 4, 6, 8, 10, 12, \ldots\}\)

### 3. \(\{x \mid x^3 - 6x^2 + 11x - 6 = 0\}\)

To describe this set, we need to find the roots of the cubic equation \(x^3 - 6x^2 + 11x - 6 = 0\).

- Factoring the cubic equation: \(x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3) = 0\).
- The roots (solutions) of this equation are \(x = 1, 2, 3\).

**Roster form**: \(\{1, 2, 3\}\)

### 4. \(\{x \mid x = \frac{1}{n}, n \in \mathbb{N}\}\)

This set includes all numbers of the form \(\frac{1}{n}\), where \(n\) is a natural number (\(\mathbb{N} = \{1, 2, 3, \ldots\}\)):

- When \(n = 1\), \(x = \frac{1}{1} = 1\)
- When \(n = 2\), \(x = \frac{1}{2}\)
- When \(n = 3\), \(x = \frac{1}{3}\), and so on.

**Roster form**: \(\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\}\)

Would you like more details or have any questions?

### Additional Questions:

1. What are natural numbers, and how are they different from whole numbers?
2. How do we solve a cubic equation with real roots?
3. What is the general form for converting a set defined by a condition into roster form?
4. Can a set have an infinite number of elements in roster form?
5. How can we find the roots of higher-order polynomials?

**Tip:** When writing sets in roster form, ensure you list all distinct elements and use ellipses (`...`) to indicate an ongoing pattern for infinite sets.

Understand how to describe sets in roster form with detailed examples from mathematics and algebra. Learn about sets defined by conditions and how to convert them into roster form. Suitable for Grades 11-12 students studying set theory and algebra.

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