Question 1194127: Given a set A = {
7
,
5
,
6
,
H
,
G
,
n
}, answer the following questions:
a. Find the number of subsets of A
Found 4 solutions by MathLover1, greenestamps, math_tutor2020, ikleyn:
Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website! Given a set A = {7,5,6,H,G,n}, answer the following questions:
recall that: The subsets of any set consists of all possible sets including its elements and the null set. Let us understand with the help of an example.
Example: Find all the subsets of set A = {1,2,3,4}
Solution: Given, A = {1,2,3,4}
Subsets =
{}
{1}, {2}, {3}, {4},
{1,2}, {1,3}, {1,4}, {2,3},{2,4}, {3,4},
{1,2,3}, {2,3,4}, {1,3,4}, {1,2,4}
{1,2,3,4}
so, in your case
a. Find the number of subsets of A
∅ | {5} | {6} | {7} | {G} | {H} | {n} | {5, 6} | {5, 7} | {5, G} | {5, H} | {5, n} | {6, 7} | {6, G} | {6, H} | {6, n} | {7, G} | {7, H} | {7, n} | {G, H} | {G, n} | {H, n} | {5, 6, 7} | {5, 6, G} | {5, 6, H} | {5, 6, n} | {5, 7, G} | {5, 7, H} | {5, 7, n} | {5, G, H} | {5, G, n} | {5, H, n} | {6, 7, G} | {6, 7, H} | {6, 7, n} | {6, G, H} | {6, G, n} | {6, H, n} | {7, G, H} | {7, G, n} | ... (total: 64)
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The response from the other tutor shows a partial list of the subsets and then, without explanation, the answer, 64.
That doesn't help the student much in determing HOW to find that the answer is 64. If there were 10, or 100, elements in the set, you wouldn't want to find the number of subsets by listing all of them.
To see how to get the answer of 64, imagine the process of forming a subset from the 6 elements in the given set. You look at each of the elements and choose whether or not to include that element in your subset. That's 2 choices -- Yes or No -- for each element. With 6 elements, the total number of different subsets you can form is the product of those numbers of choices for all 6 elements, which is 2^6 = 64.
So, in short, the number of subsets of set with n elements is 2^n.
This result can also be seen in Pascal's Triangle.
In forming a subset from a set of 6 elements, you can choose either 0, 1, 2, 3, 4, 5, or 6 of the elements. So the total number of subsets you can form is
C(6,0)+C(6,1)+C(6,2)+C(6,3)+C(6,4)+C(6,5)+C(6,6)
But those numbers are the entries in the 6th row of Pascal's Triangle; and the sum of the entries in the 6th row of Pascal's triangle is 2^6.
So again we see that the number of subsets of a set with n elements is 2^n.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Rule: If a set has n elements, then it has 2^n possible subsets.
The proof of this can be done many ways, but the simplest (in my opinion) is to imagine a row of light switches. They can be either on or off.
If we have n of these switches to represent the n elements of the set, then we have 2*2*2*2*...*2 = 2^n different on/off combos to represent the 2^n subsets.
Note: We can define "on" to mean "include this in the subset", while "off" means "exclude it from the subset".
For instance, let's say we had the set of people code named A,B,C,D
{A,B,C,D} is the original set
Now consider the string 0,1,1,0
0 = off
1 = on
We'll exclude members A and D, while including members B and C
The string 0,1,1,0 leads us to the subset {B, C} which is one of the 2^2 = 4 possible.
Having all zeros gets us the empty set with nothing inside.
Having all ones gets us the original set. Any set is a subset of itself.
With that rule in mind, we can see that there are 6 elements in the set A = {7, 5, 6, H, G, n}
This must mean there are 2^6 = 64 different subsets.
Answer by ikleyn(52781)  (Show Source):
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