Closure and the Reals

Date: 03/26/98 at 16:46:26
From: Jonnie Zobell
Subject: How you do Closure

I've tried and tried forever to get it. The problem says something 
like, "For what set of operations is the set of positive real 
integers closed?"  I have no ideal how to do this problem and ones 
like it. Please help me.

Jonnie

Date: 03/27/98 at 09:31:55
From: Doctor White
Subject: Re: How you do Closure

Jonnie:

Welcome to the exciting world of mathematics. Closure is one of the 
Field Axioms that gives many students trouble. Lets go to the zoo for 
a moment. If a cage is closed and contains lions, no matter what you 
do they all remain lions. If a giraffe ended up in the cage with the 
lions then the gate must have been open.

In mathematics, let's think about a small set of numbers like (0,1). 
Under what operations is this set closed? To see if it is not closed 
try to find an example of two or more numbers in that cage that when 
you perform an operation on them the answer is not in that cage. (you 
only need one example to show that the cage was open.)

Let's try looking at some operations on this cage of (0,1).

   Addition: pick 0 + 1 = 1 (all three are lions)
             pick 1 + 1 = 2 (oops two lions and one giraffe)
   Thus, this set is not closed under addition.

   Subtraction: pick 1 - 0 = 1 (all three are lions)
                pick 0 - 1 = -1 (oops someone left the gate open)
   Thus, this set is not closed under subtraction.

   Multiplication: pick 1 x 0 = 0  (all lions)
                   pick 1 x 1 = 1  (all lions)
                   pick 0 x 0 = 0  (all lions)
   Since there is no other pair of numbers we can choose, this 
   cage could be closed under multiplication. It is important to note 
   that you should not limit yourself to just two numbers. You could 
   have chosen three or more such as 1 x 1 x 1 x 0 = 0 (still all 
   lions). I think you can see that there is no combination of        
   numbers from this set that will not produce a lion under   
   multiplication. Thus this set is closed under multiplication.

I hope this helps you to determine the answer to your question. 
Remember that you don't have to find a lot of examples to show that a 
set is not closed; it just takes one example to show that the gate 
was left open and a giraffe joined the lions. (Hope the lions aren't 
hungry).

Come back and see us soon.

-Doctor White, The Math Forum
Check out our web site! http://mathforum.org/dr.math/