In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
events
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
is jointly or collectively exhaustive if at least one of the events must occur. For example, when rolling a
six-sided die, the events 1, 2, 3, 4, 5, and 6 balls of a single
outcome are collectively exhaustive, because they encompass the entire range of possible outcomes.
Another way to describe collectively exhaustive events is that their
union must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if
:
where S is the
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
.
Compare this to the concept of a set of
mutually exclusive events
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
. In such a set no more than one event can occur at a given time. (In some forms of mutual exclusion only one event can ever occur.) The set of all possible die rolls is both mutually exclusive and collectively exhaustive (i.e., "
MECE"). The events 1 and 6 are mutually exclusive but not collectively exhaustive. The events "even" (2,4 or 6) and "not-6" (1,2,3,4, or 5) are also collectively exhaustive but not mutually exclusive. In some forms of mutual exclusion only one event can ever occur, whether collectively exhaustive or not. For example, tossing a particular biscuit for a group of several dogs cannot be repeated, no matter which dog snaps it up.
One example of an event that is both collectively exhaustive and mutually exclusive is tossing a coin. The outcome must be either heads or tails, or p (heads or tails) = 1, so the outcomes are collectively exhaustive. When heads occurs, tails can't occur, or p (heads and tails) = 0, so the outcomes are also mutually exclusive.
History
The term "exhaustive" has been used in the literature since at least 1914. Here are a few examples:
The following appears as a footnote on page 23 of Couturat's text, ''The Algebra of Logic'' (1914):
:"As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the ''principle of exhaustion'', inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are ''exhaustive (of the universe of discourse)''." (italics added for emphasis)
In
Stephen Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
's discussion of
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
s, in ''Introduction to Metamathematics'' (1952), he uses the term "mutually exclusive" together with "exhaustive":
:"Hence, for any two cardinals M and N, the three relationships M < N, M = N and M > N are 'mutually exclusive', i.e. not more than one of them can hold. ¶ It does not appear till an advanced stage of the theory . . . whether they are '' 'exhaustive' '', i.e. whether at least one of the three must hold". (italics added for emphasis, Kleene 1952:11; original has double bars over the symbols M and N).
See also
*
Event structure In mathematics and computer science, an event structure represents a set of events, some of which can only be performed after another (there is a ''dependency'' between the events) and some of which might not be performed together (there is a ''c ...
*
MECE principle
*
Probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
*
Set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
References
Additional sources
* LCCCN: 59-12841
*{{cite book, author=Tarski, Alfred , date= 1941, edition= Reprint of 1946 2nd edition (paperback), title=Introduction to Logic and to the Methodology of Deductive Sciences, publisher= Dover Publications, Inc, location= New York, isbn=0-486-28462-X
Probability theory