To determine the number of subsets of a set, you can use the following formula:
If a set has \( n \) elements, the number of subsets \( S \) of that set is given by:
\[ S = 2^n \]
This formula arises because for each element in the set, there are two choices: include that element in the subset or do not include it.
Example Calculation
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Set with 0 elements (the empty set):
- \( n = 0 \)
- \( S = 2^0 = 1 \) (The empty set itself)
-
Set with 1 element (e.g., {a}):
- \( n = 1 \)
- \( S = 2^1 = 2 \) (The subsets are {}, {a})
-
Set with 2 elements (e.g., {a, b}):
- \( n = 2 \)
- \( S = 2^2 = 4 \) (The subsets are {}, {a}, {b}, {a, b})
-
Set with 3 elements (e.g., {a, b, c}):
- \( n = 3 \)
- \( S = 2^3 = 8 \) (The subsets are {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c})
-
Set with 4 elements (e.g., {a, b, c, d}):
- \( n = 4 \)
- \( S = 2^4 = 16 \) (The subsets are all combinations of the 4 elements)
Summary of Results
- For a set with 0 elements: 1 subset
- For a set with 1 element: 2 subsets
- For a set with 2 elements: 4 subsets
- For a set with 3 elements: 8 subsets
- For a set with 4 elements: 16 subsets
You can use this method to determine the number of subsets for any finite set of size \( n \).