disjoint

Disjoint Sets

Two sets $A$ and $B$ are disjoint if they have no elements in common, i.e., their intersection is empty:

$$A\cap B=\varnothing$$

A family of sets $(A_i{)}_{i\in I}$ is called pairwise disjoint (or mutually disjoint) if every pair of distinct sets in the set is disjoint:

$$A_i\cap A_j=\varnothing \text{for all }i\ne j\in I$$

Example

The sets $A=\{1,2,3\}$ and $B=\{4,5,6\}$ are disjoint.
The sets of even and odd integers are disjoint.
The intervals $(0,1)$ and $(1,2)$ on the real line are disjoint.

Probability of $A$ or $B$ (at least one) occuring

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

For non-disjoint events $A,B\in \Omega$ (i.e. they can co-occur), we need to subtract the intersection, so we don’t count it twice → $\cup \leftrightarrow +$ (with a correction).
For disjoint events, it simplifies:

$$(A\cap B)=\varnothing ⟹P(A\cup B)=P(A)+P(B)$$

”$A$ and $B$ happened”: “$A$ or $B$ happened”:

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