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IntervalMemberQ

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

IntervalMemberQ[interval, x]

gives True if x lies within interval, and False otherwise. IntervalMemberQ[interval, other] tests whether the interval other is wholly contained in interval.

Examples (4)

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)

2 lies in [1, 3]

In[1]:= IntervalMemberQ[Interval[{1, 3}], 2]
Out[1]= True

In[2]:= IntervalMemberQ[Interval[{1, 3}], 5]
Out[2]= False

Subset test: [2,3] inside [0,10]

In[3]:= IntervalMemberQ[Interval[{0, 10}], Interval[{2, 3}]]
Out[3]= True

Pi ~ 3.14159 is just outside

In[4]:= IntervalMemberQ[Interval[{0, 3}], Pi]
Out[4]= False

Implementation notes

Algorithm. builtin_intervalmemberq takes IntervalMemberQ[interval, x]. The first argument must be an Interval[...] (else it declines, NULL); the second is either a scalar or another Interval. Both forms reduce to endpoint comparisons via interval_endpoint_cmp, which sets an undecided flag when a comparison cannot be decided (symbolic or incomparable endpoints):

  • Scalar membership — x is a member iff for some pair [lo_i, hi_i] of the interval both lo_i <= x and x <= hi_i. If no pair contains it and every comparison was decidable, return False; if any comparison was undecidable and none matched, decline (NULL).
  • Subset test (x is itself an Interval) — every pair of x must sit inside some pair of the interval; a piece that escapes returns False, an undecidable containment declines, and all-contained returns True.

Data structures. The multi-pair Interval[{a1,b1},{a2,b2},...] is read through the interval_pair_count / interval_pair_lo / interval_pair_hi accessors shared across the interval module; endpoints are kept as exact Expr when exact.

Complexity / limits. O(n) for scalar membership over n interval pieces, O(m·n) for the subset test of an m-piece interval. Attributes: Protected. Comparisons are exact where the endpoints are exact (so IntervalMemberQ[Interval[{0, 3}], Pi] is False, Pi being about 3.14159); an endpoint whose order cannot be decided leaves the call unevaluated rather than guessing.

Attributes: Protected.

References

Notes & additional examples

Notes

IntervalMemberQ[interval, x] tests membership. With a scalar x it asks whether x lies in one of the interval's [lo, hi] pieces; with a second Interval it asks whether that interval is wholly contained in the first.

Comparisons are exact when the endpoints are exact, so IntervalMemberQ[Interval[{0, 3}], Pi] is False because Pi exceeds 3. If an endpoint ordering cannot be decided (a symbolic bound), the call is left unevaluated rather than guessing. IntervalMemberQ is Protected.