Functions

See also: Mathematical Analysis - index

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A relational graph  [;F;]  is said to be functional (a functional graph)   [;\Leftarrow :\Rightarrow;]

[;\bullet\quad\quad\quad \forall_{\ x\ y\ z}\ \left(\ (x\ y)\ (x\ z)\ \in\ F\ \ \Rightarrow\ \ y=z\ \right);]

EXAMPLE 0  The diagonal  [;\Delta_X;]  of any set  [;X;]  is a functional graph.

EXAMPLE 1  The following relational graph

[;\bullet\quad\quad\quad C_{X\,c}\ :=\ X\times \{c\};]

is functional.

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On the other hand, when set  [;A;]  is non-empty, and set  [;B;]  has more than one element, then their cartesian product  [;A\times B;]  is a relational graph which is not functional.

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A relation  [;f := (X\ Y\ F);]  is said to be a function from  [;X;]  to  [;Y;]   [;\Leftarrow :\Rightarrow;]   [;F;]  is a functional graph such that

[;\bullet\quad\quad\quad\forall_{\ x\in X}\ \exists_{\ y\in Y}\ (x\ y) \in F;]

NOTATION:   When  [;f := (X\ Y\ F);]  is a function, and  [;(x\ y)\in F;]  then we write  [;y = f(x);].  (the value  [;f(x);]  is defined uniquely because graph  [;F;]  is functional). We also write  [;f : X \rightarrow Y;], and

[;\bullet\quad\quad\quad graph(f) := F;]

Thus


[;\bullet\quad\quad\quad graph(f)\ =\ \{ (x\ f(x)) : x\in X \};]

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EXAMPLE 2 (compare with Example 0)  The identity relation

[;\bullet\quad\quad\quad I_X := (X\ X\ \Delta_X);]

is a function, called the identity function in  [;X;].  Now another way to define function  [;I_X : X \rightarrow X;]  is:

[;\bullet\quad\quad\quad \forall_{x \in X}\ I_X(x) := x;]

EXAMPLE 3 (compare with Example 1)  Let  [;c\in Y;].  The relation

[;\bullet\quad\quad\quad \gamma\ := (X\ Y\ C_{X\,c});]

is a function called the constant function on  [;X;]  (and into  [;Y;]), with constant value  [;c;].

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We will see in a next post that general functions can be constructed using more special functions, namely canonical surjections, bijections, and canonical injections. The definitions of (arbitrary) surjections, bijections, and (arbitrary) injections are provided already below.

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A function  [;f : X \rightarrow Y;]  is said to be surjective   [;\Leftrightarrow\quad \forall_{y\in Y}\exists_{x\in X}\ f(x)=y;].

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A function  [;f : X \rightarrow Y;]  is said to be injective   [;\Leftrightarrow\quad \forall_{x'\,x"\in X}\ (f(x')=f(x")\ \Rightarrow\ x'=x");].

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A function  [;f : X \rightarrow Y;]  is said to be bijective   [;\Leftrightarrow;]   [;f;]  is both surjective and injective.

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Going back to surjections, let  [;\mathbf{T} := (X\ X\ T);]  be an equivalence relation. Let  [;X/\mathbf{T};]  be the set of the equivalence classes of  [;\mathbf{T};].  Then the canonical projection is defined as a function

[;\bullet\quad\quad\quad \pi : X \rightarrow X/\mathbf{T};]

defined by equality:

[;\bullet\quad\quad\quad \forall_{x\in X}\ \pi(x) := [x] ;]

where  [;[x];] is the equivalence class of  [;x;]  with respect to the equivalence relation  [;\mathbf{T};].

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Going back to injections, let  [;A \subseteq X;].  Then the canonical embedding of  [;A;]  into  [;X;]  is defined as a function:


[;\bullet\quad\quad\quad i_A : A \rightarrow X;]

such that

[;\bullet\quad\quad\quad \forall_{x\in X}\ i_A(x) := x ;]