Learn English – 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?

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.

Best Answer

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.