2Selected Mathematical and Statistical Results
If it can't be expressed in figures, it is not science. It is opinion.
(Source: Robert Heinlein)
In this chapter we present, mostly without proof, various definitions, results, and algorithms needed in the sequel. They serve to introduce the notation and concepts of item response theory in the remaining chapters.
2.1 Points, Point Sets, and Set Operations
- A point represents any entity that can be labeled. Usually, we will label a point by a lower‐case Roman letter, such as
or 
. Two distinct points cannot have the same label, and the same point cannot have two different labels unless the second label is declared equivalent to (aliased to) the first. - A point set or, briefly, set, is a collection of points. Usually, we will represent a set by an upper‐case Roman letter, such as
. For example, the set consisting of the points 
and 
will be represented as
(2.1)The points and are called members of the set .
- The points of a set may themselves ...
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