Operation Closure Formula

$\in S$ means 'belongs to set $S$'; closure means the result of $\circ$ stays in $S$

The Formula

If a, b \in S, then a \circ b \in S

When to use: Adding two whole numbers always gives a whole number—closed under addition.

Natural numbers are closed under +: 3+5=8 Not closed under -: 3-5=-2 (not natural).

\in S means 'belongs to set S'; closure means the result of \circ stays in S

When an operation on elements of a set always produces an element in the same set.

Adding two whole numbers always gives a whole number—closed under addition.

S \text{ is closed under } \circ \iff \forall a, b \in S: a \circ b \in S

easy

Are whole numbers closed under addition? Test with $5 + 7$ and $12 + 0$. Explain what closure means.

1
Closure means: performing the operation on two members of the set always gives another member of the same set.
2
Test: $5 + 7 = 12$. Is 12 a whole number? Yes.
3
Test: $12 + 0 = 12$. Is 12 a whole number? Yes.
4
Whole numbers are closed under addition: the sum of any two whole numbers is always a whole number.

Answer

Yes — whole numbers are closed under addition

A set is closed under an operation if applying the operation to any elements of the set always produces another element in the set.

medium

Are whole numbers closed under subtraction? Provide a counterexample if not.

Concluding closure from a few examples — 2 - 1 = 1 works, but 1 - 2 = -1 breaks closure for natural numbers
Forgetting that division of integers is not closed: 7 \div 2 = 3.5, which is not an integer
Confusing closure with the idea that every operation always gives a valid answer — 5 \div 0 is undefined, which means division is not closed even on nonzero reals when 0 is included

Explains why we need to expand number systems (naturals to integers to rationals).

When an operation on elements of a set always produces an element in the same set.

Adding two whole numbers always gives a whole number—closed under addition.

\in S means 'belongs to set S'; closure means the result of \circ stays in S

Explains why we need to expand number systems (naturals to integers to rationals).

Checking closure requires testing all possible inputs, not just examples.

Before studying the Operation Closure formula, you should understand: addition, subtraction, multiplication, division.