Definition:Disjoint
Definition
A pair of properties is described as disjoint if and only if each one can't happen if the other holds.
Disjoint Sets
In the context of set theory:
Two sets $S$ and $T$ are disjoint if and only if:
- $S \cap T = \O$
That is, disjoint sets are sets such that their intersection is the empty set -- they have no elements in common.
Disjoint Events
In the context of probability theory:
Let $A$ and $B$ be events in a probability space.
Then $A$ and $B$ are disjoint if and only if:
- $A \cap B = \O$
It follows by definition of probability measure that $A$ and $B$ are disjoint if and only if:
- $\map \Pr {A \cap B} = 0$
Disjoint Permutations
In the context of permutation theory:
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\pi, \rho \in S_n$ both be permutations on $S_n$.
Then $\pi$ and $\rho$ are disjoint if and only if:
- $(1): \quad i \notin \Fix \pi \implies i \in \Fix \rho$
- $(2): \quad i \notin \Fix \rho \implies i \in \Fix \pi$
That is, each element moved by $\pi$ is fixed by $\rho$ and (equivalently) each element moved by $\rho$ is fixed by $\pi$.
That is, if and only if their supports are disjoint sets.
Also known as
Objects that are disjoint are also referred to as being mutually exclusive.
Also see
- Results about disjoint can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): mutually exclusive