Mutually exclusive

In logic, two mutually exclusive (or "mutual exclusive" according to some sources) propositions are propositions that logically cannot both be true. To say that more than two propositions are mutually exclusive may, depending on context mean that no two of them can both be true, or only that they cannot all be true. The term pairwise mutually exclusive always means no two of them can both be true.

In probability theory, events E₁, E₂, ..., E_n are said to be mutually exclusive if the occurrence of any one of them automatically implies the non-occurrence of the remaining n − 1 events. In other words, two mutually exclusive events cannot both occur.

In short, mutual exclusivity implies that at most one of the events may occur. Compare this to the concept of being collectively exhaustive, which means that at least one of the events must occur.

Examples

*A flipped coin coming up heads and the same coin coming up tails at the same time is not possible as they are mutually exclusive events.
*A student passing a test and failing it are mutually exclusive (though someone can fail a test, retake it, and then pass- or have the grade scaled).
*When rolling a six-sided die, each of the outcomes 1, 2, 3, 4, 5 and 6 are mutually exclusive and collectively exhaustive, because no more than one outcome can occur simultaneously and they encompass the entire range of possible outcomes.
*The logical operation A XOR B means A and B are mutually exclusive and cannot both be true at the same time. In order for A XOR B to be true, either A or B must be true but not both.

See also

*Excluded middle
*Collectively exhaustive
*Mutual exclusion