A set with n elements can have 2n subsets.
Here's a breakdown of why:
Each element in the original set has two possibilities when forming a subset: it's either included in the subset or it's not. Since there are n elements, and each has two choices, the total number of possible subsets is 2 2 ... 2 (n* times), which is 2n.
Example:
Let's say we have a set A = {a, b, c}. This set has 3 elements (n = 3).
Therefore, the number of subsets is 23 = 8.
The subsets are:
- {} (empty set)
- {a}
- {b}
- {c}
- {a, b}
- {a, c}
- {b, c}
- {a, b, c}
Proper Subsets:
It's also worth noting the concept of proper subsets. A proper subset of a set is a subset that is not equal to the original set itself. So, a proper subset doesn't include the set {a, b, c} in the above example. The number of proper subsets is 2n - 1. In this case, 23 - 1 = 7.
