note-math

composition (function composition)

Define proposition, parameter

• injective := . Notation
• surjective := . Notation
• bijective := injective and surjective. Notation . At this time, there exists an inverse mapping

[cardinal]

• . or there exists a bijection
• . or there exists an injection
• . or there exists a surjection

• Symmetry of injection and surjection

  or there exists a surjection <==> there exists an injection

[cardinal-always-comparable] Trichotomy of element number order or order is always comparable

[finite] := . also let

finite <==>

is a finite set ==> ( is injective or surjective <==> is bijective)

Example is an infinite set, is injective and not surjective, so not bijective

• countably infinite :=
• [uncountable] uncountable :=
• [countable] countable := i.e. finite or countably infinite

Operations that preserve countability. let , is countable. The following sets are countable

• union: ,
• product: ,

[range] . alias image of , ,

[codomain] . alias range 值域

let

[image] Image like

let

[inverse-image] Inverse image

==>

Inverse image maintains , e.g.

Image only maintains , for others

[cardinal-increase] (cf. cardinal)

Partition . Directly or according to the inverse image of the image set of the mapping ,

[quotient] quotient := or