Sketch a graph of each of the following Cartesian products in the Cartesian plane. (a) [0, 2] × [1, 3] (b) (0, 2) × (1,3] (c) [2,3] x {1} (d) {1} x [2, 3] (e) Rx (2, 4) (f) (2,4) x R (g) Rx{-1} (h) {-1}x [1, +∞0)
Sketch a graph of each of the following Cartesian products in the Cartesian plane. (a) [0, 2] × [1, 3] (b) (0, 2) × (1,3] (c) [2,3] x {1} (d) {1} x [2, 3] (e) Rx (2, 4) (f) (2,4) x R (g) Rx{-1} (h) {-1}x [1, +∞0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Transcription for Educational Website:**
The image provides instructions to sketch graphs of specific Cartesian products on the Cartesian plane. Below are the descriptions of the Cartesian products to be sketched:
(a) \([0, 2] \times [1, 3]\)
(b) \((0, 2) \times (1, 3)\)
(c) \([2, 3] \times \{1\}\)
(d) \(\{1\} \times [2, 3]\)
(e) \(\mathbb{R} \times (2, 4)\)
(f) \((2, 4) \times \mathbb{R}\)
(g) \(\mathbb{R} \times \{-1\}\)
(h) \(\{-1\} \times [1, +\infty)\)
**Explanation of Graphs:**
1. **Interval Notation:**
- Square brackets \([ \, ]\) indicate that the end values are included (closed interval).
- Parentheses \(( \, )\) indicate that the end values are not included (open interval).
2. **Graphs:**
- Each Cartesian product will be graphed on a coordinate plane.
- For example, (a) \([0, 2] \times [1, 3]\) represents a rectangle in the plane with vertices at \((0, 1), (0, 3), (2, 1),\) and \((2, 3)\).
- (b) \((0, 2) \times (1, 3)\) represents an open rectangle excluding the boundary lines.
- (c) and (d) correspond to horizontal and vertical line segments, respectively.
- (e) and (f) involve infinite strips.
- (g) represents a horizontal line across \(y = -1\).
- (h) represents a ray starting at \((-1, 1)\) and extending infinitely upward.
Each graph illustrates a different region or set of points based on the Cartesian product notation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6195f099-697f-489e-94ae-c77f1a063c08%2F293165a9-d7bd-42cc-9b09-e3a04bee8997%2Fe12a7xb_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
The image provides instructions to sketch graphs of specific Cartesian products on the Cartesian plane. Below are the descriptions of the Cartesian products to be sketched:
(a) \([0, 2] \times [1, 3]\)
(b) \((0, 2) \times (1, 3)\)
(c) \([2, 3] \times \{1\}\)
(d) \(\{1\} \times [2, 3]\)
(e) \(\mathbb{R} \times (2, 4)\)
(f) \((2, 4) \times \mathbb{R}\)
(g) \(\mathbb{R} \times \{-1\}\)
(h) \(\{-1\} \times [1, +\infty)\)
**Explanation of Graphs:**
1. **Interval Notation:**
- Square brackets \([ \, ]\) indicate that the end values are included (closed interval).
- Parentheses \(( \, )\) indicate that the end values are not included (open interval).
2. **Graphs:**
- Each Cartesian product will be graphed on a coordinate plane.
- For example, (a) \([0, 2] \times [1, 3]\) represents a rectangle in the plane with vertices at \((0, 1), (0, 3), (2, 1),\) and \((2, 3)\).
- (b) \((0, 2) \times (1, 3)\) represents an open rectangle excluding the boundary lines.
- (c) and (d) correspond to horizontal and vertical line segments, respectively.
- (e) and (f) involve infinite strips.
- (g) represents a horizontal line across \(y = -1\).
- (h) represents a ray starting at \((-1, 1)\) and extending infinitely upward.
Each graph illustrates a different region or set of points based on the Cartesian product notation.
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