Is $(-\infty, 0)$ the same size as $(0, \infty)$? A differential equations problem asked about the largest interval on which the solution was defined. The solution was defined except for $t=0$, which made me wonder whether the intervals $(-\infty, 0)$ and $(0, \infty)$ are the same size.
If these intervals are the same size, does that mean that $\mathbb{R}$ can be divided in half?
 A: You're thinking too much. Here "the largest interval on which the solution is defined" means "the largest interval containing the initial value on which the solution is defined". It has nothing to do with lengths of intervals or cardinalities or anything: if the equation has a singularity at $t=0$ and nowhere else, and $t_0<0$, then the largest interval of definition is $(-\infty,0)$. Reverse that when $t_0>0$.
The tricky thing is that it can happen that the domain of the formula for the solution is bigger than the domain of the solution itself. For instance, $y'=-y^2,y(1)=1$ has a solution $y(t)=1/t$. Although $1/t$ is defined on $(-\infty,0) \cup (0,\infty)$, the solution to the IVP is defined only on $(0,\infty)$.
A: They are the same size. Take $A,$ the set of all numbers $x \leq 0$ and $B,$ the set of all numbers $x \geq 0.$ Notice that for any element $x$ in $A,$ we can use the function $f(x) = -x$ to get to the corresponding element in $B.$ We can use the same function to go backwards. Since the sets are in 1-1 correspondence, they are equal.
However, you cannot say that $(-\infty, 0)$ is half of $(-\infty, \infty),$ because both are of infinity size.
A: In order to answer this question we need a precise definition of the size of a set.
There are several approaches to this, such as using cardinality or measure theory. We are going to stick to the first one.
In real life, sizes are made through comparisons. For example, we say that a house is 20 meters tall because its height corresponds to 20 times that of a bar of iridium which we have agreed its one meter tall.
We are going to use a similar procedure here, and define some standard sets to which we can compare the sizes of other sets.
We are going to say that two sets are of the same size if there exists a correspondence between them. A correspondence is defined as a one-to-one and onto function, which respectively mean that the function does not send two elements of its domain to the same element of the range; and that every element of its target set is assigned to an element of its domain.
Our definition agrees with our intuition in that $(0\ \infty)$ and $(-\infty\ 0)$ are the same size: consider the correspondence $f(x) = -x$.
Now comes the susprising part. It turns out that according to our definition, $(0\ \infty)$ and $\mathbb{R}$ are of the same size!
To see why this is the case, consider $f(x)=log(x)$. $f$ would be a correspondence which maps $(0\ \infty)$ to $\mathbb{R}$.
So the answer to your question is yes. $\mathbb{R}$ can be divided in half, but it turns out that each half is of the same size as the original set.
