What are Sets and Subsets?

Date: 09/06/2001 at 10:16:14
From: Andrea Gyde
Subject: What is a subset?

Dear Dr. Math,

Please tell me what a subset is.

Thank you.

Date: 09/06/2001 at 11:21:21
From: Doctor Ian
Subject: Re: What is a subset?

Hi Andrea,

It appears (from another question that you submitted) that you don't 
know what a set is, either, so I'll start from there, since there's no 
way to explain 'subsets' without talking about 'sets'.  

A set is a collection of unique objects.  For example, 
 
  {bob, carol, ted, alice}

is a set, while

  {bob, bob, carol, alice}

is not a set, because some elements are repeated.  

Note that the order of the elements is not important. Two sets are the 
same if they contain the same elements, e.g., 

  {bob, carol, ted, alice} = {alice, bob, carol, ted}

You can use set operations to manipulate sets, just as you can use 
arithmetic operators to manipulate numbers. For example, the 'union' 
operator creates a new set that contains every element that appears in 
either of the sets it operates on:

  {tom, betty} union {alice, betty, fred} = {tom, betty, alice, fred}

Note that the size of the union (4 elements) is less than the size of 
the sums of the input sets (2 and 3). This is because 'betty' appears 
in both sets.  

The 'intersection' operator creates a new set that contains every 
element that appears in _both_ of the sets it operates on:

  {tom, betty} intersect {alice, betty, fred} = {betty}
  
Note that the intersection is empty when no element appears in both 
input sets:

  {a, b, c} intersect {d, e} = {}

There is a lot more to say about sets, but this should be enough to 
give you an idea of what they are, and to suggest the kinds of things 
that can be done with them.

A _subset_ is a set that you can make by choosing none, some, or all 
of the elements of a set. For example, if I have the set

  {barney, betty, fred}

the empty set {} is a subset, which I can create by selecting none of 
the elements of the set. Similarly, the set

  {barney, betty, fred}

is a subset, which I can create by selecting all of the elements of 
the set. Note that the empty set is a subset of _every_ set. Also, a 
set is always a subset of itself. 

The most interesting subsets leave some elements out. For example, 
here are all the different subsets that I haven't mentioned so far:

  one element:    {barney}
                  {betty}
                  {fred}

  two elements:   {barney, betty}
                  {barney, fred}
                  {betty, fred}

So the subsets of the set are

  no elements:    {}

  one element:    {barney}
                  {betty}
                  {fred}

  two elements:   {barney, betty}
                  {barney, fred}
                  {betty, fred}

  all elements:   {barney, betty, fred}

Another way to think about the same thing is that you can create a 
subset, not by choosing elements to _keep_, but by choosing elements 
to get rid of:

                       {barney, betty, fred}
                   /            |             \
     {barney, betty}       {barney, fred}        {betty, fred}
        /       \             |         |           /     \
  {barney}     {betty}   {barney}     {fred}   {betty}   {fred}
     |            |         |            |        |         |
    { }          { }       { }          { }      { }       { }

All the subsets in the list also appear in the tree. The only 
difference is that instead of building up subsets starting with the 
empty set, I whittled away subsets starting with the complete set. 
     
Sets can seem a little confusing, but it's really just a formalization 
of things that you already know intuitively, and if you keep that in 
mind, you should have an easier time in your math classes.  

For example, if you have four friends in your math class (Bob, Carol, 
Ted, and Alice) and you have three friends in your English class 
(Carol, Ted, and Jane), which friends are in both classes?  You can 
probably write down the answer without too much thought: Carol and 
Ted. You've just performed a set intersection.

Which friends are in at least one class? Again, you can probably write 
down the answer easily: Bob, Carol, Ted, Alice, and Jane. Note that 
you wouldn't write Carol's name or Ted's name more than once. What 
would be the point of that? You've just performed a set union. 

Which friends are in your math class but not your English class? A 
little thought gives: Bob and Alice. You've just taken the difference 
of two sets. And you've never even _heard_ of a set difference until 
now. 

So it's really less like learning something new, and more like 
learning to translate something you already know about into a slightly 
different language. 

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/