Order as many of the following sentences as you can so that they form a logical sequence of subsets each contained in the next. Write 1,2,3,... next to the subsets in your sequence (ordered by containmnet from smallest to largest) and write 0 next to any subset not in your sequence.


  1. {3, 2, 4, 1, 5}

  2. {3, 2, 4, 1}

  3. {2, 4}

  4. {3, 2, 1}

  5. {2}

  6. {1}

  7. {4}

  8. {3, 4, 1}

  9. {3, 4}

  10. ∅

  11. {2, 4, 1, 5}

We need to order the sets from smallest to largest by containment, meaning that a smaller set should be a subset of the larger ones.

Here is the ordered sequence based on set containment:

1. \( \emptyset \) (Set 10)  
2. \( \{2\} \) (Set 5)  
3. \( \{1\} \) (Set 6)  
4. \( \{4\} \) (Set 7)  
5. \( \{2, 4\} \) (Set 3)  
6. \( \{3, 4\} \) (Set 9)  
7. \( \{3, 4, 1\} \) (Set 8)  
8. \( \{3, 2, 1\} \) (Set 4)  
9. \( \{3, 2, 4, 1\} \) (Set 2)  
10. \( \{2, 4, 1, 5\} \) (Set 11)  
11. \( \{3, 2, 4, 1, 5\} \) (Set 1)

### Final ordering:

1. ∅ (Set 10)
2. {2} (Set 5)
3. {1} (Set 6)
4. {4} (Set 7)
5. {2, 4} (Set 3)
6. {3, 4} (Set 9)
7. {3, 4, 1} (Set 8)
8. {3, 2, 1} (Set 4)
9. {3, 2, 4, 1} (Set 2)
10. {2, 4, 1, 5} (Set 11)
11. {3, 2, 4, 1, 5} (Set 1)

All sets are part of the sequence.

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Would you like more details or have any questions?

Here are five related questions:
1. What is a subset in set theory?
2. What is the definition of the empty set?
3. Can a set be a subset of itself?
4. How can we visualize the subset relations between sets using diagrams?
5. What is the largest possible subset of a set?

**Tip:** In set theory, a set is always a subset of itself. This is called the *reflexive property of sets*.

Order as many of the following sentences as you can so that they form a logical sequence of subsets each contained in the next. Write 1,2,3,... next to the subsets in your sequence (ordered by containment from smallest to largest) and write 0 next to any subset not in your sequence.

1. {3, 2, 4, 1, 5}
2. {3, 2, 4, 1}
3. {2, 4}
4. {3, 2, 1}
5. {2}
6. {1}
7. {4}
8. {3, 4, 1}
9. {3, 4}
10. ∅
11. {2, 4, 1, 5}

Learn how to order subsets based on containment, starting from the smallest set (∅) to the largest. This problem introduces key set theory concepts, including subsets, empty sets, and the reflexive property of sets. Suitable for students in Grades 9-12.

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