

A set is closed (under an operation +, , x, ÷) 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:

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 example that does NOT work
to prove closure fails. This is called a counterexample. 





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 considered to be addition, subtraction, multiplication and division, it is possible that a more creative binary operation may be created (such as Φ or any symbol) 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). Read more about "binary operations" under Algebra. 


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