Closure Property
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.
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.
Example: 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.
