K12 Math Grade 7

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  • Topic: Mathematics, Set, Subset
  • Pages : 5 (1113 words )
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  • Published : June 18, 2013
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This activity investigates how many subsets a set has.
What is a Subset?
A subset is a set contained in another set
It is like you can choose ice cream from the following flavors: {banana, chocolate, vanilla}
You could choose any one flavor {banana}, {chocolate}, or {vanilla}, Or any two flavors: {banana, chocolate}, {banana, vanilla}, or {chocolate, vanilla}, Or all three flavors (no that isn't greedy),

Or you could say "none at all thanks", which is the "empty set": {}  
Example: The set {alex, billy, casey, dale}
Has the subsets:
* {alex}
* {billy}
* etc ...
It also has the subsets:
* {alex, billy}
* {alex, casey}
* {billy, dale}
* etc ...
Also:
* {alex, billy, casey}
* {alex, billy, dale}
* etc ...
And also:
* the whole set: {alex, billy, casey, dale}
* the empty set: {}
Now let's start with the Empty Set and move on up ...
The Empty Set
How many subsets does the empty set have?
You could choose:
* the whole set: {}
* the empty set: {}
But, hang on a minute, in this case those are the same thing! So the empty set really has just 1 subset (which is itself, the empty set). It is like asking "There is nothing available, so what do you choose?" Answer "nothing". That is your only choice. Done. A Set With One Element

The set could be anything, but let's just say it is:
{apple}
How many subsets does the set {apple} have?
* the whole set: {apple}
* the empty set: {}
And that's all. You can choose the one element, or nothing.
So any set with one element will have 2 subsets.
A Set With Two Elements
Let's add another element to our example set:
{apple, banana}
How many subsets does the set {apple, banana} have?
It could have {apple}, or {banana}, and don't forget:
* the whole set: {apple, banana}
* the empty set: {}
So a set with two elements has 4 subsets.
A Set With Three Elements
How about:
{apple, banana, cherry}
OK, let's be more systematic now, and list the subsets by how many elements they have: Subsets with one element: {apple}, {banana}, {cherry}
Subsets with two elements: {apple, banana}, {apple, cherry}, {banana, cherry} And:
* the whole set: {apple, banana, cherry}
* the empty set: {}
In fact we could put it in a table:
 | List| Number of
subsets|
zero elements| {}| 1|
one element| {apple}, {banana}, {cherry} | 3|
two elements| {apple, banana}, {apple, cherry}, {banana, cherry}| 3| three elements| {apple, banana, cherry}| 1|
Total:| 8|
(Note: did you see a pattern in the numbers there?)
Sets with Four Elements (Your Turn!)
Now try to do the same for this set:
{apple, banana, cherry, date}
Here is a table for you:
 | List| Number of
subsets|
zero elements| {}|  |
one element|  |  |
two elements|  |  |
three elements|  |  |
four elements|  |  |
Total:|  |
(Note: if you did this right, there will be a pattern to the numbers.)  
Sets with Five Elements
And now:
{apple, banana, cherry, date, egg}
Here is a table for you:
 | List| Number of
subsets|
zero elements| {}|  |
one element|  |  |
two elements|  |  |
three elements|  |  |
four elements|  |  |
five elements|  |  |
Total:|  |
(Was there a pattern to the numbers?)
 
Sets with Six Elements
What about:
{apple, banana, cherry, date, egg, fudge}
OK ... we don't need to complete a table, because...
... you should be able to see a pattern by now!
Doubling
The first thing to notice is that the total number of subsets doubles each time: A set with n elements has 2n subsets
So you should be able to answer:
* How many subsets are there for a set of 6 elements? _____ * How many subsets are there for a set of 7 elements? _____ Another Pattern
Let's go back to consider how many subsets of each size there were * The empty set has just 1 subset:  1
* A set with one element has 1 subset with no elements and 1 subset...
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