A = {x|x ∈ Z
+, x = f(y) and x < 10}, where f : Z
+ → Z
+, f(y) = 2y − 1
B = {x|x ∈ R and x
2 − 7x + 12 = 0}
(a) Represent A and B by roaster method

### Problem:

Given the sets:

1. \( A = \{x \mid x \in \mathbb{Z}^+, x = f(y) \text{ and } x < 10\} \), where \( f : \mathbb{Z}^+ \to \mathbb{Z}^+ \) and \( f(y) = 2y - 1 \).
   
2. \( B = \{x \mid x \in \mathbb{R} \text{ and } x^2 - 7x + 12 = 0 \} \).

You are asked to represent both sets in the **roster method**.

---

### Solution:

#### 1. **Set A**:

- \( f(y) = 2y - 1 \), where \( y \in \mathbb{Z}^+ \) (positive integers) and \( x = f(y) < 10 \).
- Calculate values of \( f(y) \) for \( y = 1, 2, 3, \dots \) such that \( x < 10 \):

  - For \( y = 1 \), \( f(1) = 2(1) - 1 = 1 \)
  - For \( y = 2 \), \( f(2) = 2(2) - 1 = 3 \)
  - For \( y = 3 \), \( f(3) = 2(3) - 1 = 5 \)
  - For \( y = 4 \), \( f(4) = 2(4) - 1 = 7 \)
  - For \( y = 5 \), \( f(5) = 2(5) - 1 = 9 \)
  - For \( y = 6 \), \( f(6) = 2(6) - 1 = 11 \), which exceeds 10, so we stop here.

Thus, the values of \( x \) that belong to set \( A \) are:
\[
A = \{1, 3, 5, 7, 9\}
\]

---

#### 2. **Set B**:

- The equation is \( x^2 - 7x + 12 = 0 \).
- Solve the quadratic equation by factoring:

\[
x^2 - 7x + 12 = (x - 3)(x - 4) = 0
\]
\[
x = 3 \quad \text{or} \quad x = 4
\]

Thus, the values of \( x \) that belong to set \( B \) are:
\[
B = \{3, 4\}
\]

---

### Final Representation (Roster Method):

- Set \( A \): \( \{1, 3, 5, 7, 9\} \)
- Set \( B \): \( \{3, 4\} \)

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Do you want more details or have any questions?

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Here are 5 related questions for practice:
1. Find the union of sets \( A \) and \( B \).
2. Find the intersection of sets \( A \) and \( B \).
3. Find the complement of set \( A \) in \( B \).
4. If \( C = \{x \mid x \in \mathbb{Z}^+, x < 8 \} \), represent \( A \cup C \) using roster notation.
5. Solve \( f(y) = 2y - 1 \) for \( y \in \mathbb{R} \) instead of \( \mathbb{Z}^+ \).

### Tip:
When solving quadratic equations, factoring is a fast and reliable method if the equation can be easily factored.

A = {x | x ∈ Z+, x = f(y) and x < 10}, where f : Z+ → Z+, f(y) = 2y − 1
B = {x | x ∈ R and x^2 − 7x + 12 = 0}
(a) Represent A and B by roster method

Solve how to represent the sets A and B using the roster method. Set A involves a linear function f(y) = 2y - 1, and set B is derived from the quadratic equation x^2 - 7x + 12 = 0. This solution covers key algebraic concepts, including factoring quadratics and set theory.

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