# Does the definition of "mutually exclusive" include the case where neither A nor B happen?

Does the definition of "mutually exclusive" include the case where neither A nor B happen?

I ask this because I remember in logic, "xor" has to be A or B and doesn't include the case where neither A nor B happen.

Here is an example that might help explain what I want to ask.

A: seeing Superman

B: seeing Clark Kent

We can only see either Superman or Clark Kent at a time, and never see them both, and we see neither of them sometimes.

In this case, can we say A and B are mutually exclusive?

• "A and B are mutually exclusive" means only that A and B can't happen at the same time. (But I think this question would be more appropriate at e.g. english.stackexchange.com, there isn't really a philosohpy aspect here.) Apr 29, 2021 at 17:30
• Yes, it includes it. To exclude neither happening the term is "collectively exhaustive". See Wikipedia or dictionaries for these types of questions Apr 29, 2021 at 17:41

"Mutually exclusive" does not mean "A XOR B." It means "NOT (A AND B)" or equivalently "(NOT A) OR (NOT B)."

"Mutually exclusive" means either one excludes the other and is synonymous with the concept of disjointness.
In logic terms, "A and B are mutually exclusive" should be written as (A IMPLIES (NOT B)) AND (B IMPLIES (NOT A)).
If we make a truth table for this, we see its truth value is TRUE when A = B = FALSE.

If in addition to being mutually exclusive A and B exhaust the possibilities for what can happen (e.g. a coin flip can come up either A = heads or B = tails), we often say that A and B "partition", or form a partition of, the set of possibilities. But this is a strictly stronger condition than mutual exclusivity, and is not implied by mere mutual exclusivity.