I am having a hard time understanding how to prove a question such as the following
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.
a) the integers greater than 10
b) the odd negative integers
for part a) I put that it is countably infinite.
This is how I proved it: $f: \mathbb{Z}^+ \rightarrow A$, $f(x)= x+10$
$f(x)$ is one-to-one because $f(m)=f(n)$ implies $m=n$
$f(x)$ is onto because: $y=f(x), y= x+10, x=y-10$, therefore, $f(x)= f(y-10)= (y-10)+10=y$
So therefore, f is a one-to-one correspondance with the set of positive integers. Did I prove this correctly?
Now for part b) I put countably infinite and I tried to do the same:
$f: \mathbb{Z}^+ \rightarrow A, f(x)= -(2x+1)$
It is one-to-one because $f(m)=f(n)$ implies $m=n$
onto: $y=f(x), y=-(2x+1), -y-1/2=x$
This is where I got stuck. If $y=2$, we get that $x= -3/2$, which is not part of $\mathbb{Z}$? That is a real number, not an integer. Therefore, it isn't surjective because there is a value for $y$, for which $y$ does not equal $f(x)$? But I know for a fact that the answer is countably infinite, and if I think about it, it makes sense, but I am not sure why I am so stuck on wrapping my head around this part. Can someone help me?
$y = 2$ is not an odd negative integer. ;-)
Your reasoning is correct.