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.