Closed
Understand closure property in Grade 9 algebra: a set is closed under an operation if performing that operation on any two elements always produces a result within the same set.
Key Concepts
Property A set is closed under addition if the sum of any two elements from the set is also in the set.
Examples Integers: 3 + ( 5) = 2 → still an integer ✅ Real Numbers: 2.5 + ( 1.3) = 1.2 → still a real number ✅.
Explanation When a set is closed under addition, adding any two numbers from that set will never produce a number outside the set. This means no matter which two numbers you pick from the set, their sum will still belong to the same set.
Common Questions
What does it mean for a set to be 'closed' under an operation?
A set is closed under an operation if applying that operation to any two elements of the set always produces a result that is also in the set. Integers are closed under addition: any two integers added together give another integer.
Are integers closed under division?
No. Integers are not closed under division because dividing two integers can produce a non-integer result. For example, 7 ÷ 2 = 3.5, which is not an integer. A single counterexample is enough to disprove closure.
What sets are closed under multiplication and why?
Integers, rational numbers, and real numbers are all closed under multiplication because multiplying two members of each set always produces another member of that set. Natural numbers are also closed: multiplying two positive integers always gives a positive integer.