# Determine Whether Two Date Ranges Overlap

(StartA <= EndB) and (EndA >= StartB)

Proof:
Let ConditionA Mean that DateRange A Completely After DateRange B

``````_                        |---- DateRange A ------|
|---Date Range B -----|                          _
``````

(True if `StartA > EndB`)

Let ConditionB Mean that DateRange A is Completely Before DateRange B

``````|---- DateRange A -----|                        _
_                          |---Date Range B ----|
``````

(True if `EndA < StartB`)

Then Overlap exists if Neither A Nor B is true –
(If one range is neither completely after the other,
nor completely before the other,
then they must overlap.)

Now one of De Morgan’s laws says that:

`Not (A Or B)` <=> `Not A And Not B`

Which translates to: `(StartA <= EndB) and (EndA >= StartB)`

NOTE: This includes conditions where the edges overlap exactly. If you wish to exclude that,
change the `>=` operators to `>`, and `<=` to `<`

NOTE2. Thanks to @Baodad, see this blog, the actual overlap is least of:
{ `endA-startA`, `endA - startB`, `endB-startA`, `endB - startB` }

`(StartA <= EndB) and (EndA >= StartB)`
`(StartA <= EndB) and (StartB <= EndA)`

NOTE3. Thanks to @tomosius, a shorter version reads:
`DateRangesOverlap = max(start1, start2) < min(end1, end2)`
This is actually a syntactical shortcut for what is a longer implementation, which includes extra checks to verify that the start dates are on or before the endDates. Deriving this from above:

If start and end dates can be out of order, i.e., if it is possible that `startA > endA` or `startB > endB`, then you also have to check that they are in order, so that means you have to add two additional validity rules:
`(StartA <= EndB) and (StartB <= EndA) and (StartA <= EndA) and (StartB <= EndB)`
or:
`(StartA <= EndB) and (StartA <= EndA) and (StartB <= EndA) and (StartB <= EndB)`
or,
`(StartA <= Min(EndA, EndB) and (StartB <= Min(EndA, EndB))`
or:
`(Max(StartA, StartB) <= Min(EndA, EndB)`

But to implement `Min()` and `Max()`, you have to code, (using C ternary for terseness),:
`(StartA > StartB? Start A: StartB) <= (EndA < EndB? EndA: EndB)`

NOTE4. Thanks to Carl for noticing this, but another answer shows an equivalent mathematical expression for this logical expression. Because the product of any two real numbers with opposite sign is negative and with the same sign it is positive, if you convert the datetimes to fractional numbers (and most DBMSs internally use numbers to represent datetimes), the above logical expression can also be evaluated using the following mathematical expression:

``````(EndA  - StartA) * (StartB - EndB) <= 0
``````