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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:
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.
1.5 x 2.1 = 3.15
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 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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