Closure (Sets) — Definition, Formula & Examples

Closure is the property that a set has when performing a specific operation on any elements of the set always produces a result that is also in the set. For example, the integers are closed under addition because adding any two integers always gives another integer.

A set $S$ is said to be closed under a binary operation $*$ if for every $a, b \in S$ , the result $a * b \in S$ . More generally, $S$ is closed under an $n$ -ary operation $f$ if $f(a_1, a_2, \ldots, a_n) \in S$ whenever all $a_i \in S$ .

How It Works

To test whether a set is closed under an operation, you check whether every possible application of that operation to elements in the set yields a result that stays in the set. A single counterexample — one pair of elements whose result falls outside the set — is enough to prove the set is not closed. You do not need to check every pair if you can find a general argument covering all cases.

Worked Example

Problem: Determine whether the set

S = \{0, 1, 2, 3, 4, \ldots\}

(the non-negative integers) is closed under subtraction.

Pick elements from S: Choose two elements from

S

, say

a = 2

and

b = 5

a = 2,\; b = 5

Apply the operation: Compute

a - b

2 - 5 = -3

Check membership: Is

-3

S

? No — the set

S

contains only non-negative integers, and

-3

is negative.

-3 \notin S

Answer: The set of non-negative integers is not closed under subtraction, because the counterexample

2 - 5 = -3

produces a result outside the set.

Why It Matters

Closure is a foundational requirement in abstract algebra: a group, ring, or field must be closed under its operations. In linear algebra, verifying closure under addition and scalar multiplication is the first step when proving something is a subspace.

Common Mistakes

Mistake: Testing only a few examples and concluding the set is closed.

Correction: A few successful examples do not prove closure. You need a general argument that covers all elements, or a single counterexample to disprove it.