Closed (sets)

Pronunciation: /kloʊzd/ ?

A circle representing the set of integers. Inside the circle are two points labeled 4 and 7. An arrow goes from the points 4 and 7 to a small circle labeled . An arrow goes from the circle labeled  to a point inside the large circle labeled 11.
Figure 1: The set of integers is closed with respect to addition. Created with GeoGebra.
A circle representing the set of integers. Inside the circle are two points labeled 4 and 7. An arrow goes from the points 4 and 7 to a small circle labeled division. An arrow goes from the circle labeled division to a point outside of the large circle labeled four sevenths.
Figure 2: The set of integers is not closed with respect to division. Created with GeoGebra.

If a set is closed with respect to an operation, the result of that operation on any two members of the set is also a member of the set.[1] Stated mathematically:

Let S be a set that is closed with respect to the operation *. Let a ∈ S and b ∈ S. Then a * b ∈ S.

Example 1: Closure and the Set of Integers

Take the set of all integers {..., -2, -1, 0, 1, 2, ...} and the operation of addition (+). If we add any two integers, do we always get an integer? If we add 5 and 3, we get 8, which is an integer. The answer is yes, so we can say that the set of integers is closed with respect to addition.

Now consider the set of integers and multiplication. Is the set of integers closed with respect to multiplication? Can you think of any two integers that, when multiplied, give something that is not an integer? There is none, so the set of integers is closed with respect to multiplication.

Now consider the set of integers and division (÷). Are there any two integers that, when divided, do not give an integer? Try 1 and 2. 1÷2 = 0.5. 0.5 is not an integer, so we say that the set of integers is not closed with respect to division.

References

  1. closed. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-01-04). http://www.merriam-webster.com/dictionary/closed.
  2. Niven, Ivan and Zuckerman, Herbert S.. An Introduction to the Theory of Numbers, 3rd edition, pg 194-195. John Wiley & Sons, Inc., 1972. (Accessed: 2010-01-11). http://www.archive.org/stream/introductiontoth000497mbp#page/n211/mode/1up.
  3. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pg 29. Thomson, Brooks/Cole, 2005.

More Information

  • David McAdams. Closed (geometry). allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-10-28. http://www.allmathwords.org/article.aspx?lang=en&id=Closed%20(Geometry).

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Cite this article as:


Closed (sets). 2010-01-11. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/closed.html.

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Revision History


2010-01-11: Added "References" (McAdams, David.)
2009-10-29: Added figures 1 and 2. (McAdams, David.)
2009-10-28: Added examples of sets and elements such as 'addition (+)'. (McAdams, David.)
2008-06-07: Corrected spelling (McAdams, David.)
2008-03-25: Revised More Information to match current standards (McAdams, David.)
2007-07-13: Initial version (McAdams, David.)

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