ChaCha Rumba II. I. III. V.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Transcribed Image Text:**Title: Understanding Venn Diagrams in Dance Competition Heat Analysis**
**Question:**
a) Using the information from the Heat Schedule provided, complete the following Venn diagram by listing the competitors' numbers as elements within the appropriate regions based on which dances the competitor danced in these twelve heats of competition.
*Note:* Do not use ellipsis, but list each element of the set separated by a comma in numerical order in curly braces like this: { }. If the set is empty, type Empty Set without curly braces.
**Diagram Explanation:**
The Venn diagram consists of three intersecting circles representing different dance styles: ChaCha, Rumba, and Swing. These circles create eight distinct regions, labeled I to VIII, where competitors' numbers should be placed according to their participation in each dance:
- **Region I:** Represents competitors who participated only in ChaCha.
- **Region II:** Represents competitors who participated in both ChaCha and Rumba, but not in Swing.
- **Region III:** Represents competitors who participated only in Rumba.
- **Region IV:** Represents competitors who participated in both ChaCha and Swing, but not in Rumba.
- **Region V:** Represents competitors who participated in all three dances: ChaCha, Rumba, and Swing.
- **Region VI:** Represents competitors who participated in both Rumba and Swing, but not in ChaCha.
- **Region VII:** Represents competitors who participated only in Swing.
- **Region VIII:** Represents competitors who did not participate in any of the dances.
**Task:**
LIST the elements in each of the following regions in set notation:
1. Region I
2. Region II
3. Region III
4. Region IV
5. Region V
6. Region VI
7. Region VII
8. Region VIII
_Fill in the elements for each region based on the provided Heat Schedule._
Transcribed Image Text:### Ballroom Dance Competition Overview
In a recent ballroom dance competition, each couple was identified by a number worn on the leader's back. Forty couples participated, with their numbers listed as follows:
**Couple Numbers:** 109, 111, 118, 122, 123, 128, 132, 136, 145, 153, 160, 165, 171, 176, 182, 184, 188, 190, 193, 199, 202, 205, 210, 213, 217, 222, 230, 233, 236, 243, 248, 252, 255, 258, 264, 270, 273, 276, 281, 287
### Heat Schedule
The competition is organized into heats, each of which consists of a particular group of dancers performing a specific dance. Below is a segment of the Heat Schedule, showing couples assigned to twelve heats for ChaCha, Rumba, and Swing dances.
#### Heat Assignments
| **Heat 39** | **Heat 40** | **Heat 41** | **Heat 42** | **Heat 43** | **Heat 44** |
|-------------|-------------|-------------|-------------|-------------|-------------|
| **ChaCha** | **Rumba** | **Swing** | **ChaCha** | **Rumba** | **Swing** |
| 109 | 109 | 109 | 128 | 123 | 122 |
| 123 | 122 | 123 | 136 | 128 | 128 |
| 136 | 165 | 190 | 153 | 165 | 136 |
| 176 | 176 | 222 | 160 | 184 | 145 |
| 188 | 182 | 230 | 184 | 194 | 153 |
| 193 | 217 | 243 | 205 | 205 | 160 |
| 213 | 222 | 243 | 233 | 233 | 184 |
| 252 | 232 | 264 | 264 | 248
Expert Solution
Step 1
The sets without repeating will be:
CHACHA
| 109 | 128 | 111 |
| 123 | 153 | 182 |
| 136 | 160 | 202 |
| 176 | 205 | 258 |
| 188 | 233 | New joins by Heat 45 |
| 193 | 243 | |
| 213 | New joins for Heat 42 | |
| 252 | ||
| 264 | ||
| 270 |
Step 2
RUMBA
| 109 | 123 | 111 |
| 122 | 128 | 118 |
| 165 | 184 | New joins by Heat 46 |
| 176 | 199 | |
| 182 | 205 | |
| 217 | 213 | |
| 222 | 248 | |
| 233 | 255 | |
| 264 | New joins by Heat 43 | |
| 276 |
Step 3
SWING
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