Functional Notations and Terminology

The notion of a function is one of the most basic in Mathematics. A set can be identified with its characteristic function. On the other hand, functions are defined in terms of sets. For various reasons, among which historical are not the least important, mathematicians use many terms to describe essentially the same concept. Following is the list of competing terms:

  • function
  • association
  • correspondence
  • transformation
  • mapping
  • relation (multi-valued function)
  • operator
  • functional

Furthermore, there are numerous terms (vector, sequence, measure, length, volume, etc.) that are functions in disguise and their functional ancestry is seldom mentioned.

To define a function one needs three elements:

  1. a domain where the function is defined,
  2. a region from where the function draws its values, and
  3. a rule that associates points from the domain with points from the region.

For a function $f$ from a domain $X$ to a region $Y$ we use the following notation:

$f: X \rightarrow Y.$

This is the only notation that refers to all three elements of the function definition. The rule, the third element, is hinted at implicitly by the function name $f$. Two functions with the same domain and region but defined by different rules, will be distinguished by different function names. We already had one definition

Function is a correspondence $f$ between elements of a space $X$ and those of a space $Y$ such that any element $x$ of $X$ has a unique corresponding element $y$ of $Y$ which is denoted $y = f(x).$

Another way of saying that an element $y$ corresponds to an element $x$ by means of a function $f$ is $f: x\mapsto y.$ For numeric functions it's often possible to describe the rule with a formula as in $f(x) = x^{2}$ which is the same as $f: x\mapsto x^{2}.$ However, for the function which is $0$ for all rational $x$ and $1,$ otherwise, there is no formula that uses only broadly accepted math notations. Of course, if a function is used very often mathematicians may decide to standardize its name. After that point on such a named function may be used as a legitimate part of a formula.

The word association is not often used as a substitute for a function perhaps because it's judged to be more vague or fundamental than function. The word correspondence is mostly used in a set-theoretical context when we talk of a 1-1 correspondence between sets. Transformation is the term used in geometry, mapping appears in topology. Customarily, operators are functions between vector spaces, functionals are operators with $Y = \mathbb{R}.$ In case $Y = \{false, true\},$ the function is most exclusively called a predicate.

Relation is the one term that is best described in the framework of set theory. A (binary) relation $R$ between two sets $X$ and $Y$ is a subset of their direct sum: $R\subset X + Y = \{(x, y):\,x\in X \, y\in Y\}.$ We often write $x R y$ to indicate the fact that $(x, y)\in R.$ Relation $R$ is a function iff $x R y_{1}$ and $x R y_{2}$ imply $y_{1} = y_{2}.$ This way a function is identified with its graph.

If $X$ is a segment $\{1, 2 , \ldots, n\}$ of the set $\mathbb{N}$ of natural numbers then functions are called vectors and we write $f_{n}$ instead of $f(n).$ The same notation is used for sequences $(X = \mathbb{N}).$ When $X = 2^{A}$ and $Y = \mathbb{R}^{+},$ the set of positive reals, we often call a function a measure. Some measures are reasonably termed length if $A = \mathbb{R},$ area if $A = \mathbb{R}^{2},$ and volume if $A = \mathbb{R}^{3}.$

When $y = f(x),$ it is customary to say that "$y$ corresponds to $x$ by means of function $f.$" This may happen that no $y$ corresponds to two different $x$'s. If this is the case, the function $f: X \rightarrow Y,$ is said to be injective or an injection. If, for every $y\in Y$ there is an $x\in X$ to which that $y$ corresponds, function $f$ is said to be surjective, a surjection, or to be onto. A 1-1 correspondence is both injective and surjective, and vice versa. (A 1-1 correspondence is also said to be bijective or a bijection.) An injective function $f: X \rightarrow Y$ is a 1-1 correspondence between $X$ and $f(X) \subset Y.$ $(f(X)$ is the subset of all the values taken by various $x$ from $X.)$ $f$ is surjective iff $f(X) = Y.$

Functions

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