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In logic and probability theory, two propositions (or events) $a$ and $b$ are mutually exclusive if they cannot both be true (occur).

Let $R$ be a binary relation over propositions such that $(a,b)\in R$ denotes $a$ and $b$ are mutually exclusive.

I'm wondering what if $a$ and $b$ are identical.

In the common sense in mathematics, does $(a,a)\in R$ hold?

If not, what is the proper term that means "Only if $a$ and $b$ are distinct, they are mutually exclusive"?

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    I know "the common sense" hardly exists. – 2017-02-23
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    No, $(a,a)\in R$ never holds. I don't think there's an existing term for what you're describing. Maybe you could share why you're looking for one to give more context? – 2017-02-23
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    In the probability context, the definition is that the intersection of the two events is empty. See https://en.wikipedia.org/wiki/Mutual_exclusivity – 2017-02-23

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How can an event be mutually exclusive with itself?   Well, it can be empty.   That is the only case.

In probability theory, events are defined as sets of outcomes.   Two sets are disjoint, or mutually exclusive, if their intersection is empty.   Saying $A$ and $B$ are mutually exclusive means exactly: $A\cap B=\emptyset$.

Only when $A=\emptyset$ does $A\cap A=\emptyset$.   So only a non-event is mutually exclusive with itself.

( Sometimes we use the weaker definition that mutual independence means $\mathsf P(A\cap B)=0$ .   In this sense, an event can only be mutually exclusive with itself if it has zero measure, ie: it is a null set . )

In logic, propositions are held to be mutually exclusive if they cannot both be true.   In the same vein as above, this means a proposition which is mutually exclusive with itself must be false.