The number of subsets of a 5-element set is 2^5 = 32. One of those is improper, so the number of proper subsets is 31.
The number of subsets of an n-element set is 2^n. One of them is improper. One of them is empty.
Consider a 3-element set, {a, b, c}. It will have 2^3 = 8 subsets. Now consider counting in base-2 (binary). This is what counting from 0 to 7 looks like:
000
001
010
011
100
101
110
111
If you let the digits correspond to a, b, c, and you put the corresponding element of the set into the subset when the digit is 1, you have
000 corresponds to { }
001 corresponds to {c}
010 corresponds to {b}
011 corresponds to {b, c}
and so on. The last subset, {a, b, c} is improper because it contains all of the elements of the original set.
As you can see, there is a 1 to 1 correspondence between binary number from 0 to 2^n-1 and subsets of an n-element set. Thus, there are 2^n subsets of an n-element set.
The number of subsets of an n-element set is 2^n. One of them is improper. One of them is empty.
Consider a 3-element set, {a, b, c}. It will have 2^3 = 8 subsets. Now consider counting in base-2 (binary). This is what counting from 0 to 7 looks like:
000
001
010
011
100
101
110
111
If you let the digits correspond to a, b, c, and you put the corresponding element of the set into the subset when the digit is 1, you have
000 corresponds to { }
001 corresponds to {c}
010 corresponds to {b}
011 corresponds to {b, c}
and so on. The last subset, {a, b, c} is improper because it contains all of the elements of the original set.
As you can see, there is a 1 to 1 correspondence between binary number from 0 to 2^n-1 and subsets of an n-element set. Thus, there are 2^n subsets of an n-element set.
