1 2 3 4 5 6 7 ...
^ ^ ^ ^ ^ ^ ^
| | | | | | |
| | | | | | |
v v v v v v v
A 1 2 3 4 5 6 ...
The above is a visualization of the bijection between the sets (\mathbb N) and (\mathbb N \cup {A}).
Imagine that these are wooden blocks like a child plays with. Alphabet blocks. On each wooden cube we write one of the top row numbers on the block. So we have a set of wooden blocks labelled (1, 2, 3, \dots), one natural number for each block.
Now we come along and repaint the label on each block with the corresponding label from the bottom row. The block labeled (1) we repaint (A). The block labelled (2) we repaint as (1), and so forth.
Are not adding or subtracting any blocks or changing them in any way other than to label them differently.
There is a line of houses on one side of a street with addresses 101, 102, 103, 104, 105. One day the city tells them that their street is being converted to even numbers only, with the odd numbers across the street, and they’ll have to change their address.
Each resident goes down to the hardware store, buys some new decorative metal numerals, and nails the numerals to the front of their house. Now they read 202, 204, 206, 208, 210.
But the houses don’t change. Each house is identical before and after the readdressing, with the sole exception of the decorative numerals representing its address.
A bijection is nothing more than a relabeling or renaming of underlying objects that do not change.
So of course there’s a 1-1 correspondence between a given set before and after repainting. It’s the exact same set, completely unchanged except for a different label.