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The round smiley faces are a closed
set. No matter what operation is performed on round smiley faces,
another round smiley face will be created. Thus, there are always
only round smiley faces in the box.
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A set is
closed (under an operation) if and only if the operation on
two elements of the set produces another element of the set. If
an element outside the set is produced, then the operation is
not closed. |
Example: If
you multiply two real numbers, you will get another real number.
Since this process is always true, it is said that the real numbers are
"closed under the operation of multiplication". There is simply no
way to escape the set of real numbers when multiplying.
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Closure: When you combine
any two elements of the set, the result is also included in the
set. |
Example:
If you add two even numbers (from the set of even
numbers),
is the sum even?
| Checking:
|
10 + 12 = 22 |
Yes, 22 is even. |
| |
6 + 8 = 14 |
Yes, 14 is even. |
| |
2 + 100 = 102 |
Yes, 102 is even. |
Since the sum (the answer) is always
even, the set of even numbers is closed
under the operation of addition.
|
Let's check out this question.
If you divide two even numbers (from the set of even numbers), is
the quotient (the answer) even? |
| Checking: |
12 / 6 = 2 |
Yes, 2 is even. |
| |
24 / 2 = 12 |
Yes, 12 is even. |
| |
100 / 4 = 25 |
NO, 25 is not even! |
When you find even ONE example that
does not work, the set is not closed under that operation. The
even numbers are not closed under division.
Example:
 |
The elements in a binary
table are displayed horizontally and vertically outside the table
(in this table, the elements are 1, 2, 3, and 4). If the
elements inside the table are limited to the elements 1, 2, 3, and
4, the table is closed under the
indicated operation. |
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