Let's solve the problem step by step.

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.