Consider the following situations:

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 





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.





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





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 





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 nonzero 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 shown below. A number from the vertical left a number from the horizontal top, yields a number in the interior of the table. For example, 3 2 = 4.



A binary table of values is closed if the elements inside the table are limited to the elements of the set. 


This table shows operation defined 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}. 
