Suppose we have the interval


How do we remove an interval from within this interval?

For example, if we wanted to remove Interval[{-3,3}] from Interval[{-5,5}] we would get Interval[{-5,-3},{3,5}].

I looked in the mathematica documentations and searched through previous questions but could not find anything.

  • 3
    $\begingroup$ There is a function that does that in the wolfram function repository. You can use ResourceFunction["IntervalComplement"][Interval[{-5, 5}], Interval[{-3, 3}]]. $\endgroup$ Commented Aug 20, 2022 at 2:56
  • $\begingroup$ I use the wolfram website to read documentation and typing complement i found that under "Wolfram Language Documentation". I am using linux and do not have the documentation installed but maybe if you write complement in the documentation search box that function will also appear. $\endgroup$ Commented Aug 20, 2022 at 3:04
  • $\begingroup$ You can also write a custom function using IntervalMemberQ first to check whether the second interval is completely contained in the first, extract the list inside Interval using Level[Interval[{-5, 5}],{2}] and then Interval @@ Partition[Sort[{-5, 5}~Join~{-3, 3}], 2]. $\endgroup$ Commented Aug 20, 2022 at 3:13
  • 1
    $\begingroup$ Intervals are regions, RegionQ[Interval[{-5,5}]] yields True, but unfortunately RegionDifference[Interval[{-5,5}],Interval[{-3,3}]] does not return something useful. But Line works, try RegionDifference[Line[{{-5},{5}}],Line[{{-3},{3}}]], and one could translate from Interval to Line and back. $\endgroup$
    – user293787
    Commented Aug 20, 2022 at 11:01

1 Answer 1


The fact that this is not built-in to Mathematica suggests that there are some non-trivial subtleties. However, I offer the following simple implementation

IntervalComplement[Interval[{a_, b_}]] := 
 Interval[{-∞, a}, {b, ∞}]
IntervalComplement[Interval[a__List]] := 
 IntervalIntersection @@ IntervalComplement@*Interval /@ {a}
IntervalDifference[a_Interval, b_Interval] := 
 IntervalIntersection[a, IntervalComplement[b]]

Test cases

int1 = Interval[{-5, 5}];
int2 = Interval[{-3, 3}];
int3 = Interval[{-5, -3}, {3, 5}];
IntervalComplement /@ {int1, int2, int3}
(* {Interval[{-∞, -5}, {5, ∞}], 
 Interval[{-∞, -3}, {3, ∞}], 
 Interval[{-∞, -5}, {-3, 3}, {5, ∞}]} *)

For your example, we have

IntervalDifference[int1, int2] == int3
(* True *)

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