Question
List all the subsets of the given set.
StartSet hare, burro, dog EndSet
Question content area bottom
Part 1
Choose the correct answer below.
A.
empty set, {hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet
B.
{hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet, StartSet hare, burro, dog EndSet
C.
empty set, {hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet, StartSet hare, burro, dog EndSet
D.
{hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet
StartSet hare, burro, dog EndSet
Question content area bottom
Part 1
Choose the correct answer below.
A.
empty set, {hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet
B.
{hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet, StartSet hare, burro, dog EndSet
C.
empty set, {hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet, StartSet hare, burro, dog EndSet
D.
{hare}, {burro}, {dog}, StartSet hare, burro EndSet, StartSet hare, dog EndSet, StartSet burro, dog EndSet
Answers
GPT-4o mini
To find all the subsets of the set {hare, burro, dog}, we can denote the original set \( S = \{ \text{hare}, \text{burro}, \text{dog} \} \).
The total number of subsets of a set with \( n \) elements is \( 2^n \). In this case, \( n = 3 \), so the total number of subsets is \( 2^3 = 8 \).
Here are the subsets of the set \( S \):
1. The empty set: \( \emptyset \)
2. Single-element subsets:
- \( \{ \text{hare} \} \)
- \( \{ \text{burro} \} \)
- \( \{ \text{dog} \} \)
3. Two-element subsets:
- \( \{ \text{hare}, \text{burro} \} \)
- \( \{ \text{hare}, \text{dog} \} \)
- \( \{ \text{burro}, \text{dog} \} \)
4. The subset containing all elements:
- \( \{ \text{hare}, \text{burro}, \text{dog} \} \)
Putting all of this together, the complete list of subsets of the set \( S \) is:
1. \( \emptyset \)
2. \( \{ \text{hare} \} \)
3. \( \{ \text{burro} \} \)
4. \( \{ \text{dog} \} \)
5. \( \{ \text{hare}, \text{burro} \} \)
6. \( \{ \text{hare}, \text{dog} \} \)
7. \( \{ \text{burro}, \text{dog} \} \)
8. \( \{ \text{hare}, \text{burro}, \text{dog} \} \)
Now, comparing this with the given options:
- Option A does not contain the empty set and is incomplete.
- Option B does not include the empty set.
- Option C includes the empty set and all subsets listed correctly.
- Option D does not include the empty set.
Thus, the correct answer is **C**:
- \( \emptyset, \{ \text{hare} \}, \{ \text{burro} \}, \{ \text{dog} \}, \{ \text{hare}, \text{burro} \}, \{ \text{hare}, \text{dog} \}, \{ \text{burro}, \text{dog} \}, \{ \text{hare}, \text{burro}, \text{dog} \} \)
The total number of subsets of a set with \( n \) elements is \( 2^n \). In this case, \( n = 3 \), so the total number of subsets is \( 2^3 = 8 \).
Here are the subsets of the set \( S \):
1. The empty set: \( \emptyset \)
2. Single-element subsets:
- \( \{ \text{hare} \} \)
- \( \{ \text{burro} \} \)
- \( \{ \text{dog} \} \)
3. Two-element subsets:
- \( \{ \text{hare}, \text{burro} \} \)
- \( \{ \text{hare}, \text{dog} \} \)
- \( \{ \text{burro}, \text{dog} \} \)
4. The subset containing all elements:
- \( \{ \text{hare}, \text{burro}, \text{dog} \} \)
Putting all of this together, the complete list of subsets of the set \( S \) is:
1. \( \emptyset \)
2. \( \{ \text{hare} \} \)
3. \( \{ \text{burro} \} \)
4. \( \{ \text{dog} \} \)
5. \( \{ \text{hare}, \text{burro} \} \)
6. \( \{ \text{hare}, \text{dog} \} \)
7. \( \{ \text{burro}, \text{dog} \} \)
8. \( \{ \text{hare}, \text{burro}, \text{dog} \} \)
Now, comparing this with the given options:
- Option A does not contain the empty set and is incomplete.
- Option B does not include the empty set.
- Option C includes the empty set and all subsets listed correctly.
- Option D does not include the empty set.
Thus, the correct answer is **C**:
- \( \emptyset, \{ \text{hare} \}, \{ \text{burro} \}, \{ \text{dog} \}, \{ \text{hare}, \text{burro} \}, \{ \text{hare}, \text{dog} \}, \{ \text{burro}, \text{dog} \}, \{ \text{hare}, \text{burro}, \text{dog} \} \)