Example 2

Let A be a set. The identity correspondence of A (no. 3) is a function, called the identity mapping of A.

Thus with every set A there is associated a family, defined by the identity mapping of A, whose index set is A and whose set of elements is A. By abuse of language, a set is sometimes referred to as a “family”, in which case it is the family thus associated with the set in question.

A function $f$ is said to be constant if for all $x$ and $x'$ in the domain of $f$ we have $f(x)=f(x')$.

¶ Let $f$ be a mapping of a set E into E. An element $x$ of E is said to be fixed under $f$ if $f(x)=x$.