A set of propositions is "mutually exclusive and exhaustive" when exactly one of the propositions must be true. For example, of the two propositions "The sky is blue" and "It is not the case that the sky is blue", exactly one of those must be the case. Therefore, the probabilities of those propositions must sum to exactly 1.

If a set of propositions is "mutually exclusive", this states that for every two distinct propositions, the probability that both of them will be true simultaneously is zero:

$\forall i : \forall j : i \neq j ⟹ P (X_{i} \land X_{j}) = 0.$

This implies that for every two distinct propositions, the probability of their union equals the sum of their probabilities:

$P (X_{i} \lor X_{j}) = P (X_{i}) + P (X_{j}) - P (X_{j} \land X_{j}) = P (X_{i}) + P (X_{j}) .$

The "exhaustivity" condition states that the union of all propositions in $X,$ has probability $1$ (the probability of at least one $X_{i}$ happening is $1$ ):

$P (X_{1} \lor X_{2} \lor \dots \lor X_{N}) = 1.$

Therefore mutual exclusivity and exhaustivity imply that the probabilities of the propositions sum to 1:

$\sum i P (X_{i}) = 1.$

Mutually exclusive and exhaustive