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 \(\mathbb Z_5\)).