Closure Property

Date: 12/22/98 at 18:58:34
From: Phillip
Subject: Closure property

I need to know what the closure property is. I have a project and one 
of the questions is that it is closed under the operation.

Date: 12/23/98 at 19:11:40
From: Doctor White
Subject: Re: Closure property

Phillip:

There are various ways to look at the property of closure. You might 
want to look in the Dr. Math Archives for other articles and comments 
on this property.

  http://mathforum.org/dr.math/   

You can picture closure as a cage at the zoo. If a cage holds giraffes 
and the gate remains closed, then the cage will hold only giraffes. If 
the gate is open, then an elephant or two might be in with the 
giraffes.

Mathematically speaking: always start by knowing what set of numbers 
you are working with. Pick two or more of those numbers, perform an 
operation on them, and see if your result is in that set. If it is, 
then that set is closed under that operation.

Let's look at the set of even numbers. If you pick two even numbers, 
and add them, is the sum even?

     ex. 2 + 4 = 6     yes, 6 is even.

As long as the sum always ends up even, the set of even numbers is 
closed under addition.

Now let's look at a couple of even numbers and the operation of 
division. If you pick two even numbers, and divide them, is the 
quotient even?

     ex.  4/2 = 2      yes, 2 is even
     ex.  4/4 = 1      no, 1 is not even.

If you find just one example where it doesn't work, then it is not 
closed.

So the even numbers are not closed under division, but are closed under 
addition.

In conclusion: know what set you are working with, pick an operation, 
check all groups of numbers with that operation, and see if the results 
are in that set.

If you need further assistance on this matter, let us know.

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