Same Cardinality

Formal Definition

Two sets $X$ and $Y$ have the same cardinality if there exists a one-to-one function mapping $X$ onto $Y$, that is, if there exists a one-to-one correspondence between $X$ and $Y$.

Informal Definition

That is, if each element of the domain gets mapped to only one and every element of the codomain.

Example(s)

$g$:$\mathbb{R}\times\mathbb{R}$ defined by $g(x)=x$3 is both one to one and onto $\mathbb{R}$.

Non-example(s)

The function $f$ : $\mathbb{R}\times\mathbb{R}$ where $f(x)=x$ 2 is not one to one because $f(2)=f(-2)=4$ but $2 \neq -2$.Also,it is not onto $\mathbb{R}$ because the range is the proper subset of all nonnegative numbers in $\mathbb{R}$.