# Closed (Sets)

Pronunciation: /kloʊzd/ Explain Figure 1: The set of integers is closed with respect to addition. Created with GeoGebra. 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. 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 {..., -3, -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.

1. McAdams, David E.. All Math Words Dictionary, closed. 2nd Classroom edition 20150108-4799968. pg 34. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. closed. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 6/25/2018. http://www.merriam-webster.com/dictionary/closed. Buy the book
3. Niven, Ivan and Zuckerman, Herbert S.. An Introduction to the Theory of Numbers. 3rd edition. pg 194-195. www.archive.org. John Wiley & Sons, Inc.. 1972. Last Accessed 6/25/2018. http://www.archive.org/stream/introductiontoth000497mbp#page/n211/mode/1up. Buy the book
4. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pg 29. Thomson, Brooks/Cole. 2005. Last Accessed 6/25/2018. Buy the book

• David McAdams. Closed (geometry). allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/c/closed_g.html.

McAdams, David E. Closed (Sets). 4/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/closed.html.

### Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)