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Closed Set of Elements


Date: 06/30/2001 at 20:22:13
From: Craig Murai
Subject: Algebra

Can someone please explain to me the meaning of the word 'CLOSED' used 
in the following sentence that I came across in a math text?

'... set of complex numbers is CLOSED  under addition... '.

Thanks. 
Craig


Date: 07/01/2001 at 12:14:19
From: Doctor Ian
Subject: Re: Algebra

Hi Craig,

A set of elements is closed under an operation if, when you apply the 
operation to elements of the set, you always get another element of 
the set. 

For example, the whole numbers are closed under addition, because if 
you add two whole numbers, you always get another whole number - there 
is no way to get anything else. 

But the whole numbers are _not_ closed under subtraction, because you 
can subtract two whole numbers to get something that is not a whole 
number, e.g., 

  2 - 5 = -3

The integers are closed under multiplication (if you multiply two 
integers, you get another integer), but they are _not_ closed under 
division, since you can divide two integers to get a rational number 
that isn't an integer. 

The rationals, however, are closed under addition, subtraction, 
multiplication, and division.  

So the statement that 'the complex numbers are closed under addition' 
means that if you add two complex numbers together, you are guaranteed 
to get a complex number as the sum.

Does this help? 

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