definition
A set is closed (under an operation) if and only if the operation on any two elements of the set produces another element of the same set. If the operation produces even one element outside of the set, the operation is not closed.


Consider the following situations:

expin1
 The set of real numbers is closed under addition. If you add two real numbers, you will get another real number. There is no possibility of ever getting anything other than another real number.
 
5 + 12 = 17
2.5 + 3.4 = 5.9
 
3½ + 6 = 9½
closure add


   
expin2
The set of integers {... -3, -2, -1, 0, 1, 2, 3 ...} is NOT closed under division.
 
5 ÷ 2 = 2.5
Since 2.5 is not an integer, closure fails.
There are also other examples that fail.
   



one
All that is needed is ONE counterexample
to prove closure fails.


expin3
The set of real numbers is closed under multiplication. If you multiply two real numbers, you will get another real number. There is no possibility of ever getting anything other than another real number.
 
4 x 5 = 20
1.5 x 2.1 = 3.15
 
3½ x 2½ = 8¾
mr


   
expin4
The set of real numbers is NOT closed under division.
 
3 ÷ 0 = undefined
Since "undefined" is not a real number, closure fails.
 
Division by zero is the ONLY case where closure fails for real numbers.
Note: Some textbooks state that " the real numbers are closed under non-zero division" which, of course, is true. This statement, however, is not equivalent to the general statement that "the real numbers are closed under division". Always read carefully!


NOTE: While "operations" are traditionally addition, subtraction, multiplication and division, it is possible that a more creative binary operation may be created (such as Φ) which is then defined as to how it behaves when given two values. For example, Φ may be defined as a Φ b = 3a + b (multiply value in front of Φ by 3 and add value behind Φ).

Or, the situation may involve a table of values on an operation such as binaryasterisk shown below. A number from the vertical left binaryasterisk a number from the horizontal top, yields a number in the interior of the table. For example, 3 binaryasterisk 2 = 4.

expin5
A binary table of values is closed if the elements inside the table are limited to the elements of the set.
 
binarysmall
This table shows operationbinaryasteriskdefined on the set
{1, 2, 3, 4}. This operation is closed on this set since the elements inside the table are limited to only the elements in the set {1, 2, 3, 4}.

 

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