映射 Map

Refer: wiki-map

all the size condition in 2 member mapping.

单射 Injection

one-to-one;

Def:Def: x,xX,f(x)=f(x)x=x{\displaystyle \forall x,x'\in X,f(x)=f(x')\Rightarrow x=x'} OR
x,xX,xxf(x)f(x){\displaystyle \forall x,x'\in X,x\neq x'\Rightarrow f(x)\neq f(x')} 逆否命题

size(X)size(Y){size(X) \leqslant size(Y)}

满射 Surjection

Def:{Def:}
yY,xX such that y=f(x){\displaystyle \forall y\in Y,\exists x\in X{\text{ such that }}y=f(x)}

size(X)size(Y){size(X) \geqslant size(Y)}

双射 Bijection

one-to-one correspondence

yY,unique xX,such that,f(x)=y{\forall y\in Y, \exist \mathbf{unique}\ x\in X, \textnormal{such that},f(x)=y}

size(X)=size(Y){size(X) = size(Y)}