Answer:
Step-by-step explanation:
<em>Given:</em>
The set A = {5, 13, 17, 20}
<em>Question: </em>
Find the number of subsets of A
Find the number of proper subsets of A
<em>Simple solution by counting:</em>
Subset of A that has 0 element:
{∅} - 1 set
Subset of A that has 1 element:
{5}, {13}, {17}, {20} - 4 sets
Subset of A that has 2 elements:
{5, 13}, {5, 17}, {5, 20}, {13, 17}, {13, 20}, {17, 20} - 6 sets
Subset of A that has 3 elements:
{5, 13, 17}, {5, 13, 20}, {5, 17, 20}, {13, 17, 20} - 4 sets
Subset of A that has 4 elements:
{5, 13, 17, 20} - 1 set
In total, the number of subsets of A: N = 1 + 4 + 6 + 4 + 1 = 16
The number of proper subsets (all of subsets, except subset which is equal to original set A): N = 16 - 1 = 15
<u><em>Key-point:</em></u>
The counting method might be used for finding the number of subsets when the original set contains few elements.
The question is that, for a set that contains many elements, how to find out the number of subsets?
The answer is that: there is a fix formula to calculate the total number (
) of subsets of a set containing 
elements: N = 
With original set A = {5, 13, 17, 20}, there are 4 elements belonged to A.
=> Number of subsets of A: N = 
(same result as using counting method)
<em>Brief proof of formula: N = </em><em />
Each element of original set is considered in 2 status: existed or not.
If existed => fill that element in.
If not => leave empty.
For i.e.: empty subset means that all elements are selected as not existed, subset with 1 element means that all elements are selected as not existed, except 1 element, ... and so on.
=> From the point of view of a permutation problem, for each element in original set, there are 2 ways to select: existed or not. There are elements in total. => There are 
ways to select, or in other words, there are subsets.
Hope this helps!
:)
