are these sets equal? D = S / D = {1, 2, 3} / S = {1}, {1, 2}, {1, 3}, {1, 2, 3}


To determine if the sets \( D \) and \( S \) are equal, we need to compare the elements of each set.

- **Set \( D \)** is explicitly given as \( D = \{1, 2, 3\} \).

- **Set \( S \)** is written in a way that suggests it contains the following elements: \( S = \{\{1\}, \{1, 2\}, \{1, 3\}, \{1, 2, 3\}\} \).

In this context, \( S \) is a set of subsets, meaning its elements are not individual numbers but rather sets themselves. For instance:
- \(\{1\}\) is a subset containing only the number 1.
- \(\{1, 2\}\) is a subset containing the numbers 1 and 2.
- \(\{1, 3\}\) is a subset containing the numbers 1 and 3.
- \(\{1, 2, 3\}\) is a subset containing the numbers 1, 2, and 3.

On the other hand, \( D \) is a set of individual numbers \( \{1, 2, 3\} \).

### Conclusion:
The sets \( D \) and \( S \) are not equal because \( D \) contains individual numbers, while \( S \) contains subsets of numbers. The elements of \( D \) are \( 1, 2, 3 \), whereas the elements of \( S \) are the sets \( \{1\}, \{1, 2\}, \{1, 3\}, \{1, 2, 3\} \).

Would you like further clarification or have any other questions? Here are some related questions you might consider:

1. What is the difference between a set and a subset?
2. Can a set contain other sets as its elements?
3. What is the power set of a given set?
4. How do you determine if two sets are equal?
5. What would the union of \( D \) and \( S \) look like?

**Tip:** Always pay attention to the type of elements in a set—whether they are individual elements or subsets—especially when comparing sets.

Explore whether sets D = {1, 2, 3} and S = {{1}, {1, 2}, {1, 3}, {1, 2, 3}} are equal. Learn about set theory and subsets. Suitable for advanced high school students and beyond.

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