How this calculator works
A real interval includes every value between its bounds, subject to endpoint inclusion. Intersection keeps values that satisfy both interval conditions. Union keeps either condition and merges overlapping intervals. Touching intervals merge only if the contact point is included by at least one side. Left-ray and right-ray modes handle infinite endpoints. Compound mode combines any two one-sided comparisons with and or or.
Formula
A intersection B has lower bound max(a,c) and upper bound min(b,d). Inclusion requires every active boundary condition to include the endpoint.
Worked example
[0, 2] U [1, 3] is [0, 3]. The intersection of [-2, 1] and [0, 5] is [0, 1]. But (0, 1) U (1, 2) leaves a hole at 1 and cannot be merged.
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Frequently asked questions
Can an interval be a single point?
Yes. [a, a] contains a. If either side excludes a, that equal-bound interval is empty.
Can I enter infinity?
Choose a left or right ray for an infinite interval. Union and intersection modes combine finite intervals; compound mode also combines unbounded comparisons.
Why do touching open intervals stay separate?
Neither interval contains the shared endpoint, so their union has a missing point.