Surjective function

Example of overyective function (not injective).

In mathematics, a function:

f:XΔ Δ Andx f(x){displaystyle {begin{aligned}f:X fakelongrightarrow Yx fakelongmapsto f(x)end{aligned}}}}}

That's it. about and, epiyctive, suprajective, suryectiva, exhaustive, on or subyectiva if applied above all the codomain, that is, when each element of And{displaystyle Y} is the image of at least one element of X{displaystyle scriptstyle X}.

Formally,

Русский Русский and한 한 Andconsuming consuming x한 한 X:f(x)=and{displaystyle forall yin Yquad exists xin X:quad f(x)=y}

For everything and of And There exists x of Xwhich fulfills the function: f of x equals and.

Definition

One overyect function is a function whose image is equal to its codomain. Equivalently, a function f{displaystyle f} with domain X{displaystyle X} and co-domain And{displaystyle Y} it's overyective if for each and{displaystyle and} in And{displaystyle Y} There is at least one x{displaystyle x} in X{displaystyle X} such as f(x)=and{displaystyle f(x)=y}.

Symbolically

Yeah. f:X→ → And{displaystyle f:Xto Y} then it is said that f{displaystyle f} It's overyective if

Русский Русский and한 한 And,consuming consuming x한 한 X:f(x)=and{displaystyle forall ;yin Y,exists ;xin X:f(x)=y}

Notation

Sometimes to denote that a function f:X→ → And{displaystyle f:Xto Y} is overyective the notation is used:

f:X And{displaystyle f:Xtwoheadrightarrow Y}

Cardinality and Surjectivity

Given two sets A{displaystyle A} and B{displaystyle B}, among which there is an over-the-counter function f:A→ → B{displaystyle f:Ato B}, the cardinals must fulfill:

card(A)≥ ≥ card(B){displaystyle {mbox{card}}(A)geq {mbox{card}}}(B)}}

If there is also another overyective application g:B→ → A{displaystyle g:Bto A}, then it can be proved that there is a bijactive application between A{displaystyle A} and B{displaystyle B}by the Theorem of Cantor-Bernstein-Schröder.