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Mutually exclusive

Two sentences are Mutually Exclusive if and only if it is impossible for them to be true together. Thus, if one of them is true, the other must be false and vice versa. Note that it is possible for them to be both false together.

Truth Table Test for Mutually Exclusive

A truth-table can help you determine whether two sentences are mutually exclusive in the following way. After you construct a truth-table for the two sentences, look at the columns under the main connectives of the sentences you are considering. If there is no row under which both sentences are true, your sentences are mutually exclusive. This captures the idea that it is not possible for the sentences to be true at the same time. On the other hand, if there is at least one row where both sentences are true, then your sentences are not mutually exclusive because there is some possible state of affairs where the two statements are true.

Examples

The sentences $\phantom{\rule{0.333em}{0ex}}A\wedge ¬A$ and $A\vee B$ are mutually exclusive, as the following truth-table shows:

A B $\phantom{\rule{0.333em}{0ex}}A\wedge ¬A$ $A\vee B$
T T F T
T F F T
F T F T
F F F F

As a matter of fact, a contradiction, like $\phantom{\rule{0.333em}{0ex}}A\wedge ¬A$, and any other sentence will be mutually exclusive. This includes the case of two contradictions. To see this consider the truth table for the sentences $\phantom{\rule{0.333em}{0ex}}A\wedge ¬A$ and $¬\left(A\vee ¬A\right)$:

A $\phantom{\rule{0.333em}{0ex}}A\wedge ¬A$ $¬\left(A\vee ¬A\right)$
T F F
F F F

Mutually Exclusive Sentences and Probability

If $A$ and $B$ are two mutually exclusive sentences, then the probability assigned to their conjunction $A\wedge B$ is zero. That is, $P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)=0$. This is the formal way of saying that it is impossible for $A$ and $B$ to be true together (that is, it is impossible for $A\wedge B$ to be true).

Mutually exclusive sentences appear in Rule 4 of the probability calculus. The fact that $P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)=0$ for mutually exclusive $A$ and $B$ is the reason why there is no conflict between Rule 4 and Rule 6 which both apply to disjunctions.