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### Closure and the Reals

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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

Jonnie
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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/
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Associated Topics:
High School Analysis
High School Sets

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