What do you understand by, write down power set of each of the following sets {0,1,2,3,4,5, 6, 7} 

A **power set** of a given set is the set of all possible subsets of that set, including the empty set and the set itself. 

For the set \(\{0, 1, 2, 3, 4, 5, 6, 7\}\), the power set contains every possible combination of the elements of the set. If the original set has \(n\) elements, then its power set will contain \(2^n\) subsets. Since the given set has 8 elements, the power set will contain \(2^8 = 256\) subsets.

Here's a brief description of what the power set would look like for \(\{0, 1, 2, 3, 4, 5, 6, 7\}\):

1. **Empty set:** \(\emptyset\) or \(\{\}\)
2. **Single-element subsets:** \(\{0\}, \{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}, \{7\}\)
3. **Two-element subsets:** \(\{0, 1\}, \{0, 2\}, \{0, 3\}, \ldots, \{6, 7\}\)
4. **Three-element subsets:** \(\{0, 1, 2\}, \{0, 1, 3\}, \ldots, \{5, 6, 7\}\)
5. **All the way up to eight-element subset,** which is the set itself: \(\{0, 1, 2, 3, 4, 5, 6, 7\}\)

However, listing all 256 subsets would take up a lot of space, so the important thing to remember is the principle: the power set includes all possible combinations of elements from the original set.

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Would you like a more detailed explanation or have any specific questions about this concept?

Here are five related questions for practice:
1. What is the power set of \(\{a, b, c\}\)?
2. How many subsets are there in the power set of a set with 5 elements?
3. What is the formula for the number of subsets in a power set?
4. Does the power set of a set always include the empty set?
5. What is the largest subset in a power set?

**Tip:** The size of the power set grows exponentially with the number of elements in the original set!

What do you understand by, write down power set of each of the following sets {0,1,2,3,4,5, 6, 7}

Understand the concept of the power set and explore all 256 subsets for the set {0,1,2,3,4,5,6,7}. This problem involves set theory and combinatorics, with a detailed breakdown of subsets and the formula used to calculate the power set.

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