Function

Definition

$\forall x\in X,\forall y,z\in Y:((x,y)\in f\wedge (x,z)\in f)\Rightarrow y=z$ where $X,Y$ are sets and $f:X\to Y$

A relation $f:X\to Y$ satisfies following conditions

• $\forall x\in X:\exists y\in Y\text{ s.t. }(x,y)\in f$
• $(x,y)\in f\wedge (x,z)\in f\Rightarrow y=z$

$(x,y)\in f\iff y=f(x)$

$X$ is a Domain of $f$ $Y$ is a Codomain of $f$ $\{f(x)|x\in X\}$ is an Image of a function $f$

Properties

Image and Preimage

Image

$y=f(x)$ is an Image of an element $x$ under $f$ $f(A)=\{f(x)|x\in A\subset X\}$ is an Image of a subset $A$ under $f$

Preimage

$x=f^{-1}(y)$ is a preimage of an element $y$ under $f$ $f^{-1}(B)=\{f^{-1}(x)|x\in B\subset Y\}$ is a preimage of a subset $B$ under $f$

Restriction and Extension

Restriction

$\forall x\in S:f{|}_S(x)=f(x):S\to Y$ where $f:X\to Y$ is a function, $S\subset X$

$f{|}_S(x):S\to Y$ is a restriction of $f$ to $S$

Extension

$f=g{|}_S(x)$

The $g$ is an extension of a function $f$ when $f$ is a restriction of $g$

Injective, Surjective, Bijective Function

Injective

Definition

$\forall a,b\in X:f(a)=f(b)\Rightarrow a=b$

Maps distinct elements of its Domain to distinct elements

Facts

Injective function is an example of an Monomorphism

Link to original

Surjective

Definition

$\forall y\in Y,\exists x\in X,f(x)=y⟺\mathrm{Cod}(f)=Y$ A function is surjective if every element in the function’s codomain is the Image of at least one element of its Domain

Facts

Surjective function is an example of an Epimorphism

If a function $f:X\to Y$ is surjective, then $\forall U\subset Y,f(f^{-1}(U))=U$

Link to original

Bijective

Definition

A function that is both Injective and Surjective

Every element in the Codomain of the function is mapped to by exactly one element in the Domain of the function

Facts

Bijective function is an example of an Isomomorphism

Link to original

Composition

Let $f:X\to Y$ and $g:Y\to Z$ where the Domain of $g$ is the Codomain of $f$ $\forall x:(g\circ f)(x)=g(f(x))$

Classes

• Identity Function
• Constant Function
• Inverse Function
• Inclusion Function
• Indicator Function
• Choice Function

Facts

Even if the function expression is the same, if the Domain is different, is a different function.

e.g. Let $f(x)=x^2$. If $\mathrm{Dom}(f)=\mathbb{R}$, then $f:\mathbb{R}\to {\mathbb{R}}_0^{+}$. On the other hand, if $\mathrm{Dom}(f)=\mathbb{C}$, then $f:\mathbb{C}\to \mathbb{R}$

$g\circ f\ne f\circ g$

Associative Property: $(h\circ g)\circ f=h\circ (g\circ f)$

$f^{-1}\circ f=I_X$

$f\circ f^{-1}=I_Y$

If $f,g$ both are Injective, then $g\circ f$ is also Injective

If $f,g$ both are Surjective, then $g\circ f$ is also Surjective

If $g\circ f$ is Injective, then $f$ is Injective If $g\circ f$ is Surjective, then $g$ is Surjective

$f(\varnothing )=\varnothing$

$\forall x\in X:f(\{x\})=\{f(x)\}$

$f^{-1}(f(A))=A\iff f$ is Injective

$f(f^{-1}(B))\iff f$ is Surjective

$f(\bigcup _{i\in I}{A}_i)=\bigcup _{i\in I}f(A_i)$

$f(\bigcap _{i\in I}{A}_i)\subset \bigcap _{i\in I}f(A_i)$

If $f$ is a Injective, then $f(\bigcap _{i\in I}{A}_i)=\bigcap _{i\in I}f(A_i)$