To determine the number of subsets of a set, you can use the formula \(2^n\), where \(n\) is the number of elements in the set. This is because each element can either be included in a subset or not, leading to \(2\) choices (include or exclude) for each of the \(n\) elements.
Here’s how you can determine the number of subsets for various examples:
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Empty Set:
- Set: \({}\)
- Number of elements (\(n\)): 0
- Number of subsets: \(2^0 = 1\)
-
Set with 1 Element:
- Set: \({a}\)
- Number of elements (\(n\)): 1
- Number of subsets: \(2^1 = 2\) (subsets: \({}, {a}\))
-
Set with 2 Elements:
- Set: \({a, b}\)
- Number of elements (\(n\)): 2
- Number of subsets: \(2^2 = 4\) (subsets: \({}, {a}, {b}, {a, b}\))
-
Set with 3 Elements:
- Set: \({a, b, c}\)
- Number of elements (\(n\)): 3
- Number of subsets: \(2^3 = 8\) (subsets: \({}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}\))
-
Set with 4 Elements:
- Set: \({a, b, c, d}\)
- Number of elements (\(n\)): 4
- Number of subsets: \(2^4 = 16\)
You can apply the same reasoning to any set; simply find the number of elements and compute \(2^n\) for the number of subsets. If you have specific sets in mind, feel free to share, and I can help calculate!