Countability

Finding One to One Correspondence

Find a one-to-one correspondence from the set of positive integers to the set of all integers (positive, negative, or zero).
\[\mathbb{N} \in \> \> \{ \> \> 1, \> \> \> \> 2, \> \> \> 3, \> \> \> 4, \> \> \> 5, \> \> \> 6, \> \> \> \ 7 \}\] \[\> \> \> \> \> \> \> \> \downarrow \> \> \> \> \downarrow \> \> \> \> \> \downarrow \> \> \> \> \> \downarrow \> \> \> \> \> \downarrow \> \> \> \> \> \downarrow \> \> \> \> \> \> \downarrow\] \[\mathbb{Z} \in \{-3, -2, -1,\> \> \> 0,\> \> \> 1,\> \> \> 2, \> \> 3\}\]

Find a one-to-one correspondence from the set A={(x,y)^2:(x, y)=1} to the set B={(x,y)^2: x^2+y^2=1}.

Let S:={(x,y)^2: x^2+(y-1)^2=1 y2}. Find a one-to-one correspondence from S to the set of all real numbers.