A finite set with n elements has 2n subsets.
Let's break this down:
Understanding Subsets
A subset is a set formed from the elements of another set. For example, if we have the set A = {1, 2}, the subsets of A are:
- {} (the empty set)
- {1}
- {2}
- {1, 2}
Why 2n?
The formula 2n arises because for each element in the set, we have two choices when forming a subset: either include it or exclude it. Since these choices are independent for each of the n elements, we multiply the number of choices together: 2 2 ... 2 (n* times), which is 2n.
Examples
Here are some examples to illustrate the concept:
- Set with 0 elements (Empty Set): If a set has 0 elements (the empty set, denoted as {} or ∅), it has only one subset: the empty set itself. 20 = 1.
- Set with 1 element: Consider the set {a}. Its subsets are {} and {a}. 21 = 2.
- Set with 2 elements: Consider the set {a, b}. Its subsets are {}, {a}, {b}, and {a, b}. 22 = 4.
- Set with 3 elements: Consider the set {a, b, c}. Its subsets are {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c}. 23 = 8.
Formula Verification
We can summarize the relationship between the number of elements in a set and the number of subsets it has in the table below:
| Number of Elements (n) | Subsets | Number of Subsets (2n) |
|---|---|---|
| 0 | {} | 1 |
| 1 | {}, {a} | 2 |
| 2 | {}, {a}, {b}, {a, b} | 4 |
| 3 | {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} | 8 |
Conclusion
Therefore, a finite set containing n elements will always have 2n subsets, arising from the binary choice of including or excluding each element in the original set when forming the subsets.
