As we see in the two examples above, two events are disjoint because they have contradictory conditions to occur. Disjoint events are also known as mutually exclusive events because the occurrence of one event «excludes» the occurrence of the other event. A disjointed union can mean one of two things. The easiest way to unite disjoint sets. [11] However, if two or more sets are not already disjointed, their disjoint union can be formed by modifying the sets to be disjoined before the union of the modified sets is formed. [12] For example, two sets can be disjoint by replacing each element with an ordered pair of the element and a binary value indicating whether it belongs to the first or second set. [13] Similarly, for families with more than two sets, each element can be replaced by an ordered pair of the element and the index of the set it contains. [14] Event A and Event B would be disjointed because they cannot occur at the same time. The coin cannot land on the head and number. What are disjunctive events? Disjointed events cannot occur at the same time. In other words, they are mutually exclusive. Formally, events A and B are disjoint if their intersection is zero: P(A∩B) = 0. You will sometimes see this written as follows: P (A and B) = 0.
The two terms are equivalent. Two events are said to be disjointed (or mutually exclusive) if they cannot both occur at the same time. That is, the probability that both occur at the same time is zero. If a collection contains at least two sentences, the condition that the collection is disjoint implies that the intersection of the entire collection is empty. However, a collection of sets can have an empty intersection without being disjointed. While a collection of less than two sets is trivially disjointed because there are no pairs to compare, the intersection of a collection of a set is equal to that set, which may not be empty. [2] For example, the three sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not disjointed. In fact, there are not two disjointed sets in this collection. The empty family of sets is also disjointed in pairs. [6] For non-disjoint events A and B, the probability of A or B occurring is P(A) plus P(B) minus the probability that A and B will occur simultaneously (avoid double counting). Two sets are called almost disjoint sets when their intersection is small in one direction.
For example, it can be said that two infinite sets, whose intersection is a finite set, are almost disjointed. [3] As already mentioned, if two events are disjoint, then the probability that both occur at the same time is zero. For families, the concept of pair disjunction or mutual disjunction is sometimes defined in a subtly different way by allowing repeated identical members: the family is disjointed in pairs if A i ∩ A j = ∅ {displaystyle A_{i}cap A_{j}=varnothing } whenever A i ≠ A j {displaystyle A_{i}neq A_{j}} (the two different sets of the family are disjointed). [2] For example, the collection of sets is { {0, 1, 2}, {3, 4, 5}, {6, 7, 8},. } disjoint, as well as the set { {…, −2, 0, 2, 4, …}, {…, −3, −1, 1, 3, 5} } of the two parity classes of integers; the family ( { n + 2 k ∣ k ∈ Z } ) n ∈ { 0 , 1 , . , 9 } {displaystyle ({n+2kmid kin mathbb {Z} })_{nin {0,1,ldots ,9}}} with 10 members is not disjoint (because the classes of even and odd numbers are present five times each), but it is disjointed in pairs according to this definition (since only then do you get a non-empty intersection of two members, if both are of the same class). In topology, there are different terms of separate sets with stricter conditions than disjunction. For example, two groups may be considered distinct if they have disjunctive closures or disjointed neighborhoods. Similarly, in a metric space, positively separated sets are sets separated by a non-zero distance.
[4] When two events are inconsistent/mutually exclusive and cover all possible (exhaustive) outcomes, they are called complementary events. For example, getting heads or tails in a coin throw are two complementary events because they are mutually exclusive and comprehensive (these are the only two options of a coin throw). For disjunctive events A and B, the probability of A or B occurring is simply the probability of A plus the probability of B. Note that there is no overlap between the two sampling spaces. Therefore, events A and B are unrelated events because they cannot occur at the same time. Unrelated events and independent events are different. Events are considered disjointed if they never occur at the same time; These are also known as mutually exclusive events. Events are considered independent if they have nothing to do with each other.
If either unrelated event is to occur, the events are complementary events. Two sets A and B are disjoint exactly when their intersection A ∩ B {displaystyle Acap B} is the empty set. [1] It follows from this definition that any set is disjoint from the empty set and that the empty set is the only set separated from itself. [5] By definition, disjointed events cannot occur at the same time. A synonym for this term is mutually exclusive. If two events are disjointed, then they would not overlap at all in a Venn diagram: a partition of a set X is a collection of mutually disjointed non-empty sets whose union is X. [8] Each partition can be described equivalently by an equivalence relation, a binary relationship that describes whether two elements belong to the same set in the partition. [8] Disjoint data structures[9] and partition refinement[10] are two computational techniques for efficiently managing partitions in a set that are subject to Union operations that merge two sets, or refinement operations that divide a set into two. This definition of disjoint sets can be extended to a family of sets ( A i ), i ∈ i {displaystyle left(A_{i}right)_{iin I}}: The family is disjoint in pairs or disjoint to each other if A i ∩ A j = ∅ {displaystyle A_{i}cap A_{j}=varnothing } each time I ≠ j {displaystyle ineq j}. Alternatively, some authors use the term disjoint to refer to this term as well. Non-disjoint events, on the other hand, can occur simultaneously. These are often represented visually by a Venn diagram, like the one below.
In this diagram, there is no overlap between event A and event B. These two events never happen together, so they are disjointed events. Written in probability notation, events A and B are disjoint when their intersection is zero. This can be written as follows: Disjoint events are disjoint or disconnected. Another way to look at disjointed events is that they don`t have common outcomes. This definition may be easier to understand if you think otherwise: overlapping events have one or more outcomes in common. In the Venn diagram representation of events A and B that are not disjoint, we have two overlapping circles, or in other words, connecting, indicating that the probability that events A and B will occur at the same time is non-zero. It is therefore a number between 0 and 1. Assuming you have discovered that your events are disjoint (with the above definition), you can find the probabilities by adding them: P(A or B) = P(A) + P(B) Which can also be rewritten as: P(A∪B) = P(A) + P(B) Some examples of events that are not disjoint are: In the representation of the Venn diagram, We represent each event by the circles, if events A and B are disjointed, we get two circles that do not touch each other. A Helly family is a system of sets in which the only subfamilies with empty intersections are those that are disjointed in pairs. For example, the closed intervals of the real numbers form a Helly family: if a family of closed intervals has an empty intersection and is minimal (that is, no subfamily of the family has an empty intersection), it must be disjoint in pairs. [7] In mathematics, two sets are called disjoint sets if they do not have a common element.