Help Alice

You can actually match the natural and even numbers up.

$\begin{aligned} \textbf{For example} : & \color{midnightblue}{1 \iff 2} \\ & \color{#E81990}{2 \iff 4} \\ & \color{#D61F06}{3 \iff 6} \\ & \cdots \end{aligned}$

where the case is that for any natural number $n$ it can be matched up with even number $2n$ . To illustrate this I will show you.

$\begin{aligned} \textbf{See:} \qquad & {\color{#69047E}{n \qquad \iff 2n}} \qquad \\ & {\color{#EC7300}{n+1 \iff 2n+2}} \\ & {\color{#20A900}{ \cdots \quad \iff \cdots}} \end{aligned}$

A bijective function is a one-to one (injective) function and an onto (surjective) function. A one-to-one function is a function between two sets where every element in the first set has an image in the second set; also, every single element in the first set is related to one and only one image of the second set. An onto function is a function where every element in the first set has an image in the second set. If we took the set of even numbers, we assign to every element in that set an image in the natural numbers set and make those one-to-one too as indicated in the second image.

The cardinality of a set is the number of elements in the set. Since we are able to construct a bijective relation from the set of even numbers to the set of natural numbers, we can conclude that these two sets have the same cardinality.

Remark Images courtesy Wikipedia

Since there exist a well defined map which is one-one and onto i.e bijective from the set of natural numbers to the set of even numbers. Hence the cardinality of even numbers is equal to the cardinality of natural numbers .