Events

An event is a measurable set of outcomes. It occurs exactly when the realized outcome belongs to that set.

Use the following stipulated sample space for the formal examples:

\Omega_4 \equiv\{\textit{he},\textit{him},\textit{they},\textit{them}\}.

The observation process selects one case-coded pronoun token from a population restricted to these four forms. Within this toy representation, he and him are coded singular, they and them plural, he and they nominative, and him and them accusative. These codings define the example; they are not claims about every use of these surface strings.

Defining the number and case events

Define the plural event by

P\equiv\{\textit{they},\textit{them}\}

and the accusative event by

A\equiv\{\textit{him},\textit{them}\}.

The outcome them belongs to both P and A. Thus one outcome can make both events occur.

Intersection

The intersection contains the outcomes shared by two events:

P\cap A=\{\textit{them}\}.

Thus P\cap A occurs exactly when the selected token is both plural and accusative under the stipulated coding.

Union

The union contains the outcomes belonging to at least one of two events:

P\cup A =\{\textit{him},\textit{they},\textit{them}\}.

The connective is inclusive. The outcome them belongs to the union because it belongs to both events.

Complement

The complement is always relative to the declared sample space:

P^c \equiv\Omega_4\setminus P =\{\textit{he},\textit{him}\}.

Thus changing \Omega can change P^c even when the members of P remain fixed.

Event membership

For the outcome them,

\textit{them}\in P, \qquad \textit{them}\in A, \qquad \textit{them}\in P\cap A.

For the outcome him,

\textit{him}\notin P, \qquad \textit{him}\in A, \qquad \textit{him}\in P\cup A.

The limiting events

The empty event \varnothing contains no outcomes. The full event \Omega_4 contains every represented outcome. Both are events because both are subsets of \Omega_4.

The outcome determines whether events can co-occur

Under this model, one outcome is one pronoun token. The nominative and accusative events are thus disjoint. If one outcome were instead an entire sentence containing several pronouns, the events “contains a nominative pronoun” and “contains an accusative pronoun” could occur together.

Mutual exclusivity follows from the event sets and the unit represented by one outcome. It does not follow from the labels alone.

Code

omega <- c("he", "him", "they", "them")
plural <- c("they", "them")
accusative <- c("him", "them")

intersect(plural, accusative)
union(plural, accusative)
setdiff(omega, plural)

Check your understanding

Define the nominative event N and verify that N\cap A=\varnothing.
List P\cap A, P\cup A, and A^c.
Explain why them can make two events occur without counting as two outcomes.
Give a different outcome representation under which nominative and accusative events can co-occur.

--- title: "Events" --- An [**event**](https://bruno.nicenboim.me/bayescogsci/ch-intro.html) is a measurable set of outcomes. It occurs exactly when the realized outcome belongs to that set. Use the following stipulated sample space for the formal examples: $$ \Omega_4 \equiv\{\textit{he},\textit{him},\textit{they},\textit{them}\}. $$ The observation process selects one case-coded pronoun token from a population restricted to these four forms. Within this toy representation, *he* and *him* are coded singular, *they* and *them* plural, *he* and *they* nominative, and *him* and *them* accusative. These codings define the example; they are not claims about every use of these surface strings. ## Defining the number and case events Define the plural event by $$ P\equiv\{\textit{they},\textit{them}\} $$ and the accusative event by $$ A\equiv\{\textit{him},\textit{them}\}. $$ The outcome *them* belongs to both $P$ and $A$. Thus one outcome can make both events occur. ## Intersection The intersection contains the outcomes shared by two events: $$ P\cap A=\{\textit{them}\}. $$ Thus $P\cap A$ occurs exactly when the selected token is both plural and accusative under the stipulated coding. ## Union The union contains the outcomes belonging to at least one of two events: $$ P\cup A =\{\textit{him},\textit{they},\textit{them}\}. $$ The connective is inclusive. The outcome *them* belongs to the union because it belongs to both events. ## Complement The complement is always relative to the declared sample space: $$ P^c \equiv\Omega_4\setminus P =\{\textit{he},\textit{him}\}. $$ Thus changing $\Omega$ can change $P^c$ even when the members of $P$ remain fixed. ## Event membership For the outcome *them*, $$ \textit{them}\in P, \qquad \textit{them}\in A, \qquad \textit{them}\in P\cap A. $$ For the outcome *him*, $$ \textit{him}\notin P, \qquad \textit{him}\in A, \qquad \textit{him}\in P\cup A. $$ ## The limiting events The empty event $\varnothing$ contains no outcomes. The full event $\Omega_4$ contains every represented outcome. Both are events because both are subsets of $\Omega_4$. ## The outcome determines whether events can co-occur Under this model, one outcome is one pronoun token. The nominative and accusative events are thus disjoint. If one outcome were instead an entire sentence containing several pronouns, the events “contains a nominative pronoun” and “contains an accusative pronoun” could occur together. Mutual exclusivity follows from the event sets and the unit represented by one outcome. It does not follow from the labels alone. ```{r} #| eval: false omega <- c("he", "him", "they", "them") plural <- c("they", "them") accusative <- c("him", "them") intersect(plural, accusative) union(plural, accusative) setdiff(omega, plural) ``` ## Check your understanding 1. Define the nominative event $N$ and verify that $N\cap A=\varnothing$. 2. List $P\cap A$, $P\cup A$, and $A^c$. 3. Explain why *them* can make two events occur without counting as two outcomes. 4. Give a different outcome representation under which nominative and accusative events can co-occur.