# Opposite (sort of) of mutually exclusive?

Paint cannot be red and blue as these are mutually exclusive. It cannot be night and day as these are mutually exclusive.

However unlike paint, which can be neither red nor green (for example: blue), it's always night or day. It can impossibly be not night and not day.

Mutually exclusive indicates the impossibility of terms being true at the same time.

Is there a similar expression for the impossibility of terms being false at the same time?

(edit) Note that the expression or phrase I'm looking for, e.g. "these things are xxx" would be equal to saying that their opposites are mutually exclusive.

This is fundamentally different than saying "these things are not mutually exclusive" or "these things are mutually inclusive" (which both mean the things can co-exist, but don't rule out the possibility of none existing).

Saying things are mutually exclusive means at most one can occur. What I'm looking for should indicate that at least one must occur.

• I know it wasn't the point of the question, but it can indeed be neither night not day, it can be twilight, dusk, dimpsey or gloaming. Twilight even has subcategories :) en.wikipedia.org/wiki/Dusk#/media/… Oct 18, 2016 at 10:55
• @Spagirl OK, for sake of argument, instead of night or day, let's use even or odd numbers :) Oct 18, 2016 at 10:57
• Could you give as an example of how you want to use this? The context can make a big difference here. Offhand, I would say a value is required or mandatory. Oct 18, 2016 at 11:12

In logic and mathematics, two terms that cover your intended meanings are mutually exclusive and collectively exhaustive.

As you already know, to say that two or more events are mutually exclusive means that no two of those events can be true at the same time. You've already listed a few examples, so I won't repeat them.

In contrast, to say that a set of events is collectively exhaustive means that at least one of the events must occur. Among the integers, odd and even are collectively exhaustive; prime and composite are not, since 1 is neither prime nor composite. Night and day are (debate-ably) collectively exhaustive, since the time at your location must be either night or day. Red, blue, green, and yellow are not collectively exhaustive, since many other colors are possible.

Finally, a set of events can be both mutually exclusive and collectively exhaustive. In this case, we say that the events partition the underlying possible outcomes. Night and Day partition time, Odd and Even partition the integers, Land and Water partition the surface of the earth.

A fairly verbose phrase is it's either one or the other

one of two persons or things but not the other

It's similar to mutally exclusive, but with the added implication that it must be one of the options and not something else.

• I think "mutually inclusive" means things can occur simultaneously (in this sense it's the opposite of "mutually exclusive") but they can also occur independently (i.e. one, or some, and possibly but not necessarily all). However I don't think "mutually inclusive" implies the IMpossibility of none occurring. That's what I'm looking for. Put differently: things being mutually exclusive means at most one of them can occur. Now I'm looking for something that means at least one of them must occur. Oct 18, 2016 at 10:56
• So in essence, you're not actually looking for the opposite of mutually exclusive. I'd remove that analogy from the question in order to clear up what you mean, as it otherwise confuses the question.
– SGR
Oct 18, 2016 at 10:58