# 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.