Explanation:
A set is "closed" for a particular operation if performing that operation on members of the set always gives a member of the set.
closed example
For example, the set {0, 1} is closed for multiplication:
0 × 0 = 0
0 × 1 = 0
1 × 1 = 1 . . . all result values are members of the set
Performing multiplication on any two members of the set gives a member of the set.
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not closed example
The same set {0, 1} is not closed for addition:
0 +0 = 0
0 +1 = 1
1 +1 = 2 . . . not a member of the set
Performing addition on any two members of the set does not always give a member of the set.
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Additional comment
Sometimes you are asked to demonstrate closure of a particular set using a particular example. As we see with addition, above, some examples may seem to demonstrate closure, while another example may prove the set is not closed. In short, you can demonstrate that the set is closed for a particular operation on a particular example, but that does not demonstrate closure in general.
