Functions: Summary of Notations

Two sets $X$ and $Y$ may be related in numerous ways:

• membership relation:

  • belongs-to: $X \in Y$

  • does-not-belong-to: $X \not\in Y$

• inclusion relation:

  • is-subset-of: $X\subseteq Y$

  • is-not-subset-of: $X \not\subseteq Y$

  • is-proper-subset-of: $X\subset Y$

  • is-not-proper-subset-of: $X \not\subset Y$

• equivalence relation:

  • equal-to: $X \subseteq Y \land Y\subseteq X$

• total relation: $X \times Y$

• cardinality: $|X| = n$ (n is number of elements)

• powerset:

  • powerset of X: $\mathcal{P}(X)$

  • cardinality of the powerset: $|\mathcal{P}(X)| = 2^{|X|} = 2^n$

• overlapping: $X \cup Y \not= \varnothing$

• disjoint: $X \cup Y = \varnothing$

Set operations:

• union: $X \cup Y$

• intersection: $X \cap Y$

• difference: $X \setminus Y$

• complement: $\bar{A}$

The set of all functions, $F$, between two sets, $X$ and $Y$, is a proper subset of the set of all relations, $R$, between the two sets.

The set of relations between two sets is a subset (total relation is equal to the dot product) of the Cartesian product between the two sets.

$F \subset R \subseteq X \times Y$

A univalent or single-valued relation is a relation where every element of domain is related to a single element of codomain:

$\forall x\ \forall y_1\ \forall y_2\ \big[\ (x,y_1 \in R) \land (x,y_2 \in R) \to\ y_1=y_2\ \big]$

Univalent relations may be identified to functions whose domain is a subset of X.

A left-total relation is a relation such that

$\forall x\in X, \exists y\in Y:\;(x,y)\in R$

Formal functions may be strictly identified to relations that are both univalent and left total. Violating the left-totality is similar to giving a convenient encompassing set instead of the true domain, as explained above.

• function relation: $f (x)=y$

• function may be uniquely represented by its graph, which is a set of all the ordered pairs $(x,y)\in R$, where $y=f(x)$

• functions or maps i.e. mapping: $x\mapsto f(x)$, $f: x\mapsto y$

Various properties of functions and also the functional composition may be reformulated in the language of relations. For example, a function is injective if the converse relation

$\displaystyle R^{\text{T}} \subseteq (Y \times X)$ is univalent, where the converse relation is defined as $\displaystyle R^{\text{T}} = \{(y,x),(x,y) \in R\}$

Let $f:\mathbb{R} \to \mathbb{R}$ be the function defined by the equation $f(x) = x^2$.

The notation, $y=f(x)$, ("y equals f of x") means that the pair $(x, y)$ belongs to the set of pairs defining the function $f$. If $X$ is the domain of $f$, the set of pairs defining the function is: $\{(x,f(x))\colon x\in X\}$

To explicitly express domain $X$ and the codomain $Y$ of a function $f$, the arrow notation is often used:

$f:X\to Y\quad$

"the function f from X to Y". Or:

$X \stackrel{f} \to Y$

"the function f maps x to f (x)"

This is often used in relation with the arrow notation for elements, often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:

$x\mapsto f(x)$

Index notation is often used instead of functional notation; instead of writing $f (x)$, one writes: $f_{x}$

Image and preimage

Let $f: X\to Y$, then the image by $f$ of an element $x$ of the domain $X$ is $f(x)$. If $A$ is any subset of $X$ then the image of $A$ by $f$, denoted $f(A)$ is the subset of the codomain $Y$ consisting of all images of elements of $A$, that is, $f(A)=\{f(x)\mid x\in A\}$

The image of $f$ is the image of the whole domain, that is $f(X)$, also called the range of $f$.

The inverse image or preimage by $f$ of a subset $B$ of the codomain $Y$ is the subset of the domain $X$ consisting of all elements of $X$ whose images belong to $B$. It is denoted by $f^{-1}(B)$. That is,

$f^{-1}(B)=\{x\in X\mid f(x)\in B\}$

For example, the preimage of $\{4,9\}$ under the square function is the set $\{−3,−2,2,3\}$.

Injective

An injective or one-to-one function, or injection, is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain; every element of the function's codomain is the image of at most one element of its domain.

A function $f:X\to Y$ is injective or one-to-one if for every $x \in X$, there exists at most one $y \in Y$ such that $f(x)=y$. A function $f$ is injective if $x\neq y$ implies $f(x)\neq f(y)$.

$x\neq y \Rightarrow f(x)\neq f(y)$.

The function $f : X\to Y$ is injective if $f(x_1)\neq f(x_2)$ for any two distinct elements, $x_1, x_2 \in X$.

Surjective

A function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. The function f may map one or more elements of X to the same element of Y, so y need not be unique.

A function f:A→B is surjective (onto) if the image of f equals its range.

Equivalently, for every b∈B, there exists some a∈A such that f(a)=b. This means that for any y in B, there exists some x in A such that y=f(x).

Bijective

A function is bijective or one-to-one correspondent iff f is both injective and surjective.

A bijection is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements.

In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.

Inverse

The inverse of a one-to-one corresponding function f:A→B, is the function g:B→A, holding the following property −

f(x)=y⇔g(y)=x The function f is called invertible, if its inverse function g exists.

Example A Function f:Z→Z,f(x)=x+5, is invertible since it has the inverse function g:Z→Z,g(x)=x−5.

A Function f:Z→Z,f(x)=x2 is not invertiable since this is not one-to-one as (−x)2=x2.

Composition

Functions $f:X\to Y$ and $g:Y\to Z$ can be composed to give a composition $g\circ f$, equal to $g(f(x))$, only if the codomain of $f$ is defined as the domain of $g$.

Composition properties

• Composition is associative, $h\circ g\circ f=(h\circ g)\circ f=h\circ (g\circ f)$

• Composition is not commutative

• If $f$ and $g$ are one-to-one functions, their composition is as well.

• If $f$ and $g$ are onto functions, their composition is as well.

A function space is a set of functions between domain and codomain.

A constant function is a function whose output value is the same for every input value.

An identity function (also called an identity relation, identity map or identity transformation) is a function that always returns the same value that was used as its argument.

A linear map (also called a linear mapping, linear transformation, soemtimes also linear function) is a mapping $V \to W$ between two modules (including vector spaces) that preserves the operations of addition and scalar multiplication. An important special case is when $V = W$, in which case the map is called a linear operator, or an endomorphism of $V$. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not.

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7.

A rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

An algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division and raising to a fractional power.

An analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others.

The smoothness of a function is a property measured by the number of derivatives it has that are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain.

A continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function.

A measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.

Restriction of a function $f$ is a new function $f\vert _{A}$ obtained by choosing a smaller domain $A$ for the original function $f$. The notation $f{\upharpoonright _{A}}$ is also used.

Properties

Classes/properties

• constant: output is the same for every input.

• identity: output is the same as input.

• Linear

• Polynomial

• Rational

• Algebraic

• Analytic

• Smooth

• Continuous

• Measurable

• Injective

• Surjective

• Bijective

Constructions

• Restriction

• Composition

• lambda, λ

• Inverse

Generalizations

• Partial

• Multivalued

• Implicit

References

https://en.wikipedia.org/wiki/Function_(mathematics)