Equality of cardinality

Comparison of cardinality


$\exists f,g (f,g \text{ are injective})\small  \Rightarrow\small \exists h(h \text{ is bijective}) $


$\small |A|\leq |B|\land \small |B|\leq |A|\Rightarrow |A|=|B| $


$\small [n]:= \begin{cases}\{1,2,...,n\} (n \geq 1)\\   \phi \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (n=0)\end{cases}$


$A$ is a countable set: $|A|=|\mathbb{N}|$

$A$ is an uncountable set: $A$ is not countable set


$A$ is a finite set: $ \exists n_1\in \mathbb{N}_0(|A|=|[n_1]|)$ \\Also $|A| :=♯ A=n_1$