Question 3. Let A = {x ∈ R |² ≥ 4}.and B = {y ∈ Z | |x-1|< 2}.
(a) Describe A and B in terms of intervals and individual elements of the real line.
(b) Describe A x B geometrically as a subset of R x R, and illustrate with a diagram.

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### **Given:**
- \( A = \{ x \in \mathbb{R} \ | \ x^2 \geq 4 \} \)
- \( B = \{ y \in \mathbb{Z} \ | \ |x - 1| < 2 \} \)

### **Part (a): Describe A and B in terms of intervals and individual elements of the real line.**

#### **Step 1: Describe \( A \) in terms of intervals.**

The set \( A \) is defined by the condition \( x^2 \geq 4 \).

- \( x^2 \geq 4 \) means \( x \geq 2 \) or \( x \leq -2 \).
- This can be rewritten as two separate intervals:
  \[
  A = (-\infty, -2] \cup [2, \infty)
  \]

So, \( A \) consists of all real numbers \( x \) that are either less than or equal to \(-2\) or greater than or equal to \(2\).

#### **Step 2: Describe \( B \) in terms of individual elements.**

The set \( B \) is defined by the condition \( |x - 1| < 2 \).

- \( |x - 1| < 2 \) means:
  \[
  -2 < x - 1 < 2
  \]
  Adding \( 1 \) to all parts of the inequality:
  \[
  -1 < x < 3
  \]
- Since \( B \) consists of integers (\( y \in \mathbb{Z} \)), we take the integer values of \( x \) within this interval.

So, the set \( B \) is:
\[
B = \{0, 1, 2\}
\]

### **Part (b): Describe \( A \times B \) geometrically as a subset of \( \mathbb{R} \times \mathbb{R} \), and illustrate with a diagram.**

#### **Step 3: Describe \( A \times B \) as a subset of \( \mathbb{R} \times \mathbb{R} \).**

The Cartesian product \( A \times B \) represents all pairs \( (x, y) \) where \( x \in A \) and \( y \in B \).

- \( A = (-\infty, -2] \cup [2, \infty) \)
- \( B = \{0, 1, 2\} \)

Thus, the Cartesian product \( A \times B \) consists of all points \( (x, y) \) where:
- \( x \) is in \( (-\infty, -2] \cup [2, \infty) \)
- \( y \) is one of \( 0 \), \( 1 \), or \( 2 \).

Geometrically, this means that for each \( y \) in \( \{0, 1, 2\} \), the \( x \) values span the intervals \( (-\infty, -2] \) and \( [2, \infty) \).

#### **Step 4: Illustrate \( A \times B \) with a diagram.**

To visualize \( A \times B \), imagine the real line \( \mathbb{R} \) for \( x \) and the discrete points \( y = 0 \), \( y = 1 \), and \( y = 2 \).

The graph will consist of three horizontal strips corresponding to \( y = 0 \), \( y = 1 \), and \( y = 2 \), where each strip extends horizontally along the \( x \)-axis over the intervals \( (-\infty, -2] \cup [2, \infty) \).

This gives us a series of horizontal lines where:

- The first line (for \( y = 0 \)) spans the intervals \( (-\infty, -2] \) and \( [2, \infty) \).
- The second line (for \( y = 1 \)) spans the same intervals.
- The third line (for \( y = 2 \)) also spans the same intervals.

Would you like to see the diagram or have any more questions? 

### **Relative Questions:**
1. How does the Cartesian product differ when dealing with finite vs. infinite sets?
2. What happens to the Cartesian product \( A \times B \) if we change \( B \) to \( B = \{0\} \)?
3. How can we represent a Cartesian product in higher dimensions, such as \( \mathbb{R} \times \mathbb{R} \times \mathbb{R} \)?
4. What are the implications of \( A \times B \) if both sets \( A \) and \( B \) are continuous intervals?
5. How does the inclusion of non-integer values in \( B \) affect the geometric representation?

### **Tip:**
When dealing with Cartesian products, visualizing each set individually and then combining them can greatly simplify understanding the geometric interpretation.

Explore the mathematical concepts of sets, intervals, and Cartesian products through a detailed explanation of the sets A and B. Learn how to describe sets in terms of intervals and individual elements of the real line. Visualize the geometric representation of A × B in ℝ × ℝ with diagrams and comprehensive examples.

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