Complement of a Set

Date: 08/14/2003 at 14:34:23
From: Monica
Subject: Universal Sets...What are they?

What is a universal set? I see it in my math book, but I don't 
understand how to figure out a problem with this in it. EXAMPLE: The 
universal set is {natural numbers from 1-10}. Set J is {even numbers 
from 2-10. What is the complement of set J?

Date: 08/15/2003 at 12:03:35
From: Doctor Ian
Subject: Re: Universal Sets...What are they?

Hi Monica,

In everyday speech, when we say that something doesn't exist anywhere
'in the universe', we mean that it doesn't exist, period, because if
something isn't 'in the universe', it isn't anywhere. There isn't
anywhere else for it to be.

In math, we make up our own little universes. For example, if I say
that, for purposes of a given problem, the universe is the positive
integers, then as long as I'm working on the problem, no other numbers
exist for me.  I can't use numbers like 1.5, or the square root of 2,
or -17.  

In your case, your universe is the set of natural numbers from 1 to 
10, 

  {1, 2, 3,4, 5, 6, 7, 8, 9, 10}

and set J is

  {2, 4, 6, 8, 10}

The 'complement' of a set is everything in the universe that ISN'T in
the set.  For example, if our universe is the set

  {1, 2, 3, 4, 5}

and set A is 

  {2, 3, 4}

then the complement of A is 

  {1, 5}

If the universe is all whole numbers, then the complement of A is

  {0, 1, 5, 6, 7, 8, 9, ...}

that is, all the whole numbers EXCEPT 2, 3, and 4.  If the universe is 

  {2, 3, 4}

then the complement of A is the empty set, {} - because there is
nothing in the universe that isn't in A.  

Note that for a given set, the contents of its complement will depend
on what's been declared as the universe.  

Does this help? 

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