Question

List all the subsets of the given set.
StartSet f,k,w EndSet
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Part 1
Choose the correct answer below.
A.
​{f​}, ​{k​}, ​{w​}, ​{f​,k​}, ​{f​,w​}, ​{k​,w​}
B.
​{f​}, ​{k​}, ​{w​}, ​{f​,k​}, ​{f​,w​}, ​{k​,w​}, ​{f​,k​,w​}
C.
empty set​, ​{f​}, ​{k​}, ​{w​}, ​{f​,k​}, ​{f​,w​}, ​{k​,w​}
D.
empty set​, ​{f​}, ​{k​}, ​{w​}, ​{f​,k​}, ​{f​,w​}, ​{k​,w​}, ​{f​,k​,w​}

Answer 1

To find all the subsets of the set \( \{f, k, w\} \), we need to include the empty set and all possible combinations of the elements in the set.

The subsets of the set \( \{f, k, w\} \) are:

1. The empty set: \( \emptyset \)
2. Single element sets: \( \{f\}, \{k\}, \{w\} \)
3. Two-element sets: \( \{f, k\}, \{f, w\}, \{k, w\} \)
4. The set itself: \( \{f, k, w\} \)

Combining all these subsets, we get:

- \( \emptyset \)
- \( \{f\} \)
- \( \{k\} \)
- \( \{w\} \)
- \( \{f, k\} \)
- \( \{f, w\} \)
- \( \{k, w\} \)
- \( \{f, k, w\} \)

Now looking at the options provided:

- Option A: This option does not include the empty set nor \( \{f, k, w\} \).
- Option B: This option does not include the empty set.
- Option C: This option does not include \( \{f, k, w\} \).
- Option D: This option includes all subsets, including the empty set and \( \{f, k, w\} \).

Therefore, the correct answer is:

**D. empty set, {f}, {k}, {w}, {f, k}, {f, w}, {k, w}, {f, k, w}**.