# Closure

A set $$S$$ is closed under an operation $$f$$ if, whenever $$f$$ is fed elements of $$S$$, it produces another element of $$S$$. For example, if $$f$$ is a trinary operation (i.e., a function of three arguments) then “$S$ is closed under $$f$$” means “if $$x, y, z \in S$$ then $$f(x, y, z) \in S$$”.

For example, the set $$\mathbb Z$$ is closed under addition (because adding two integers yields another integer), but the set $$\mathbb Z_5 = \{0, 1, 2, 3, 4, 5\}$$ is not (because $$1 + 5$$ is not in $$\mathbb Z_5$$).

Parents:

• Mathematics

Mathematics is the study of numbers and other ideal objects that can be described by axioms.