Closed Set of ElementsDate: 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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