1. Draw a rough picture of each set:
a) A = [0, 4]
b) A × A, where A is as in part (a)
c) B = {(x, y) ∈ R
2
: x ≥ 4}
d) BC, where B is as in part (c)
e) D = {(x, y) ∈ R
2
: x
2 + y
2 ≤ 25}
f) B ∩ D, where B and D are as above

{"size":"1024x1024","prompt":"A series of rough sketches representing mathematical sets: \n\n1. A = [0, 4], a horizontal line segment on the x-axis from 0 to 4.\n2. A × A, a square in the xy-plane with vertices at (0,0), (0,4), (4,0), and (4,4).\n3. B = {(x, y) ∈ R² : x ≥ 4}, a shaded half-plane on the right side of the line x=4 in the xy-plane.\n4. Bᶜ, the complement of B, which is the shaded half-plane on the left side of the line x=4 in the xy-plane.\n5. D = {(x, y) ∈ R² : x² + y² ≤ 25}, a shaded circle with a radius of 5 centered at the origin.\n6. B ∩ D, the intersection of sets B and D, which is the shaded region where the half-plane B and the circle D overlap, specifically the part of the circle on or to the right of x=4."}I am unable to draw the rough sketches at the moment, but I can describe how you can sketch each of these sets:

1. **A = [0, 4]**:
   - Draw a number line (x-axis).
   - Mark points 0 and 4 on the line.
   - Shade the segment between 0 and 4, including both endpoints.

2. **A × A**:
   - In a coordinate plane (xy-plane), plot the points (0,0), (0,4), (4,0), and (4,4).
   - Connect these points to form a square.
   - Shade the entire square. This represents all points \((x, y)\) where both \(x\) and \(y\) are in the interval [0, 4].

3. **B = \{(x, y) \in \mathbb{R}^2 : x \geq 4\}**:
   - Draw a vertical line at \(x = 4\) on the xy-plane.
   - Shade the region to the right of this line. This represents all points where \(x \geq 4\).

4. **Bᶜ** (complement of B):
   - Using the same coordinate plane as in part (c), shade the region to the left of the line \(x = 4\). This represents all points where \(x < 4\).

5. **D = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 25\}**:
   - Draw a circle centered at the origin (0,0) with a radius of 5.
   - Shade the entire area inside the circle. This represents all points \((x, y)\) that satisfy the equation \(x^2 + y^2 \leq 25\).

6. **B ∩ D** (intersection of B and D):
   - Use the same coordinate plane with the circle from part (e) and the vertical line from part (c).
   - The intersection is the region within the circle that is also to the right of the line \(x = 4\).
   - Shade this overlapping area.

Would you like more details on any specific part or help with another topic?

Here are 5 related questions you might consider:
1. How would the set \( A \times A \) change if \( A = [-2, 3] \)?
2. How do you find the area of the region represented by \( B \cap D \)?
3. What happens to the graph of \( D \) if the inequality changes to \( x^2 + y^2 < 25 \)?
4. How does the complement \( B^C \) change if \( B = \{(x, y) \in \mathbb{R}^2 : x \leq 2\} \)?
5. What is the union of sets \( B \) and \( D \)?

**Tip**: Understanding the geometric representation of sets in a coordinate plane can greatly help in visualizing solutions to problems involving inequalities.

Explore the visual representation of mathematical sets including intervals, Cartesian products, inequalities, and geometric shapes. Learn how to sketch sets like A = [0, 4], A × A, B = {(x, y) ∈ R² : x ≥ 4}, Bᶜ, D = {(x, y) ∈ R² : x² + y² ≤ 25}, and B ∩ D with step-by-step instructions and explanations.

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