We often want to predict one of several outcomes.

A classifer is a
predictor whose codomain is a finite set.
In this case, we call the codomain the
label set and we call
its elements classes (or
labels,
categories).
We call the *prediction* of a classifier
on an input a
classification.

If the set of labels has two elements, then
we call the classifier a binary
classifier (or two-way
classifier, two-class
classifier, boolean
classifier).
In the case that there are $k$ labels, we
call the classifier a $k$-way
classifier (or $k$-class
classifier, multi-class
classifier).
The second term is meant to indicate, not that
the classifier assigns to each point several
classes, but that the classification decision is
made *between* several classes.

Let $A$ be a set of inputs and let $B$ be a set of labels. Define $B = \set{0, 1}$ (or $\set{-1,1}$, {False, True}, {Negative, Positive}). Then $B$ is finite with two elements and $f: A \to B$ is a binary classifier with labels $0$ and $1$.

If the case $B = $ $\{$No, Maybe, Yes$\}$, we call $f: A \to B$ a three-way classifier. Other examples for $B$ include a list of languages, the set of English words in some dictionary, or the set of $m!$ possible orders of $m$ horses in a race.

When dealing with a finite set of $k$ arbitrary objects, it is often convenient to take associate the objects with the first $k$ positive integers and take the set $B = \set{1, \dots , k}$ as the set of labels; here $k \in \N $.

Following our terminology, but speaking of processes, some authors refer to the application of inductors for these special cases as binary classification and multi-class classification. Or they speak of classification and classification problems. Roughly speaking, a classifier classifies all inputs into categories.

Alternatively, some authors (especially in the statistics literature) refer to a classifier as a discriminator and reference discrimination problems.