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...