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When something can be A or B but not both, A and B are mutually exclusive. As sets, A and B would be disjoint.

When everything must be A or B—that is, a thing may be A or B or both, but not neither—what are A and B called? A and B need not be mutually exclusive, but not-A and not-B definitely are.

Here are some examples.

Somebody took cookies from the cookie jar! It could have been Sally or Timmy, but let's not forget that they might be in cahoots; it being Sally or Timmy is [as described above], after all.

If you want to get technical, heads and tails results on a coin are not [as described above], because you missed the very tiny chance that the coin lands on its side.

Clearly, a signed graph cannot at once have a blocknode and two disjoint odd circuits—the two conditions are mutually exclusive. Ideally, they are also [as described above], but as we will show, this is not the case.

To be clear, I'm not looking for mathematical terminology, but something a fluent English speaker could recognize.

  • They are the only two options - maybe binary options (although this has a different meaning in the stock market.) – Jim Aug 20 '15 at 3:03
  • Note though, that I could perform my 5th backup on the 90th day. – Jim Aug 20 '15 at 3:04
  • @Jim I feel option carries the connotation that you must chose one, and further that you could abstain from choosing altogether. I am seeking to describe the opposite: you can have one or the other or both, but not neither. – algorithmshark Aug 20 '15 at 3:35
  • The truth table for this function is 0 1 1 0, that is ≠ aka not equal. So maybe you could call the choice a trade-off or call the options opposites. Not perfect yet, but reducing this to its essentials (≠) seems to lead in promising directions. – Dan Bron Aug 20 '15 at 10:02
  • @DanBron Incorrect, I am looking for logical OR. – algorithmshark Aug 20 '15 at 15:52
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I would label A and B as "mutually/collectively/jointly exhaustive events". It's a standard math jargon if you go and read about a concept called Venn Diagram. Hope this solves your problem.

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This gets sticky, but I suppose you're talking about inclusivity. In your first example, either inclusive or, more commonly, all-inclusive, works:

The money-back policy on the backup software is valid for up to 90 days or 5 backups. The two conditions are all-inclusive.

For your second example, just inclusive seems to fit the case (you tell me--I'm not familiar with the discourse domain):

Clearly, a signed graph cannot at once have a blocknode and two disjoint odd circuits—the two conditions are mutually exclusive. Ideally, they are also inclusive, but as we will show, this is not the case.

  • While the definition of inclusive seems to cover the case, I haven't ever seen or heard it used like this. If nothing better comes up, this may have to do. – algorithmshark Aug 20 '15 at 8:44
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And/or is quite common, ordinary speech included. You could tell someone on the phone: "The money-back policy on the backup software is valid for up to 90 days and or 5 backups." A fluent English speaker would have no problem with it.

  • "Clearly, ... the two conditions are mutually exclusive. Ideally, they are also and/or, but as we will show, this is not the case." This doesn't parse quite right. I'm looking for a phrase that would fit both examples naturally. – algorithmshark Aug 20 '15 at 8:38

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