Math / Discrete

QuestionList all subsets of the set $\{\mathrm{D}, \mathrm{O}, \mathrm{G}\}$ using proper notation

Studdy Solution

STEP 1

1. A subset is any combination of elements from a set, including the empty set and the set itself.
2. The set $\{\mathrm{D}, \mathrm{O}, \mathrm{G}\}$ has 3 elements.
3. The number of subsets of a set with $n$ elements is $2^n$ .

STEP 2

1. Determine the total number of subsets.
2. List subsets of different sizes.
3. Verify the total number of subsets.

STEP 3

Determine the total number of subsets for the set $\{\mathrm{D}, \mathrm{O}, \mathrm{G}\}$ .
The formula for the number of subsets of a set with $n$ elements is $2^n$ . Here, $n = 3$ .
$2^3 = 8$
So, there are 8 subsets.

STEP 4

List subsets of different sizes.
Subsets of size 0 (the empty set): $\emptyset$
Subsets of size 1 (single elements): $\{\mathrm{D}\}, \{\mathrm{O}\}, \{\mathrm{G}\}$
Subsets of size 2 (pairs of elements): $\{\mathrm{D}, \mathrm{O}\}, \{\mathrm{D}, \mathrm{G}\}, \{\mathrm{O}, \mathrm{G}\}$
Subsets of size 3 (the set itself): $\{\mathrm{D}, \mathrm{O}, \mathrm{G}\}$

STEP 5

Verify the total number of subsets listed.
Counting all subsets: - 1 subset of size 0 - 3 subsets of size 1 - 3 subsets of size 2 - 1 subset of size 3
Total: $1 + 3 + 3 + 1 = 8$
This matches the expected number of subsets, $2^3 = 8$ .
The subsets of the set $\{\mathrm{D}, \mathrm{O}, \mathrm{G}\}$ are: $\emptyset, \{\mathrm{D}\}, \{\mathrm{O}\}, \{\mathrm{G}\}, \{\mathrm{D}, \mathrm{O}\}, \{\mathrm{D}, \mathrm{G}\}, \{\mathrm{O}, \mathrm{G}\}, \{\mathrm{D}, \mathrm{O}, \mathrm{G}\}$

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