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How do I calculate the number of subsets?
Given a set A with cardinality n, there are 2โฟ subsets, and 2โฟ โ 1 proper subsets. Here's why:
- Consider a set with one element, {a}. There are two subsets: โ and {a}.
- Now consider a set with two elements, {a, b}. There are four subsets: โ , {a}, {b}, and {a, b}.
- This pattern goes on for any n, to deliver 2โฟ.
- However, for the number of proper subsets, we subtract one subset (representing the original set, A): 2โฟ โ 1.
What is a subset of a set?
Let A and B be two sets. We say that A is a subset of B if every element of A is also an element of B. In other words, A consists of some (possibly all) of the elements of B but doesn't have any elements that B doesn't have. If A is a subset of B, we can also say that B is a superset of A.
What is a proper subset?
A is a proper subset of B if A is a subset of B and A isn't equal to B. In other words, A has some but not all of the elements of B, and A doesn't have any elements that don't belong to B.
We can also say that B is a proper superset of A.
Pick out the number $2$ so you are left with the
set $\{1,3,4,...,n\}$. Now you construct all subsets of
this set and add a $2$ to each of these subsets.
As $\{1,3,4,...,n\}$ is a set of $n-1$ elements
it has in total $2^{n-1}$ subsets and by your
construction you have in total $2^{n-1}$ subsets
of $\{1,2,3,...,n\}$ that contains a $2$.
Hint: there is a one-to-one correspondence between the subsets of $\{1,..,n\}$ which contain $n$ and the subsets of $\{1,...,n-1\}$