**Exercise 1.** We want to find a one-to-one correspondence between
the points on a large circle and the points on a small circle. The small
circle is drawn inside the large circle.

Pick a point A inside the large circle. If x is any point on the small circle, consider the line that passes through A and x. This line will intersect the larger circle in a point x'. For the one-to-one correspondence, we just match up x with x'.

**Exercise 2.** We want to find a one-to-one correspondence between
the "half-open" interval (0,1], which include the point 1, and the
open interval (0,1), which does not include 1.

We know that 1 in (0,1] has to match up with some point in (0,1). Match 1 up with 1/2. But then we can't match 1/2 in (0,1] to 1/2 in (0,1), since the 1/2 in (0,1) already is already matched up with 1. So match 1/2 in (0,1] to 1/3 in (0,1). Then, match 1/3 in (0,1] to 1/4 in (0,1), 1/4 in (0,1] to 1/4 in (0,1), and so on.

So far, we have a one to one correspondence between the points 1, 1/2, 1/3, 1/4, 1/5, ... in (0,1] and the points 1/2, 1/3, 1/4, 1/5, 1/6, ... in (0,1). We can extend this to a one-to-one correspondence on all of (0,1] by matching every other point x in (0,1] with the same point x in (0,1). And this gives a one-to-one correspondence between (0,1] and (0,1).

**Exercise 3.** Now, we want a one-to-one correspondence between the
closed interval [-1,1] and the open interval (-1,1). This is similar to
Exercise 2, but we now have two extra points in [-1,1] to take care of.

Just as before, match the points 1, 1/2, 1/3, ... in [-1,1] up with 1/2, 1/3, 1/4, ... in (-1,1). And match -1, -1/2, -1/3, ... in [-1,1] up with -1/2, -1/3, -1/4, ... in (-1,1). Every other point x in [-1,1] is simply matched up to the same x in (-1,1).

**Exercise 4.** We have to find a one-to-one correspondence between
a "closed" disk with its boundary and an "open" disk without its boundary. In this case, we
have an infinite number of extra points to deal with. However, we can deal with
them all at once as Exercise 1 shows. We are looking at "unit disks" with radius 1.

The boundary of the closed unit disk is a circle of radius 1. By Exercise 1, we can match up the points on the boundary with the points on the circle of radius 2 in the open unit disk. Then, map the points on the circle of radius 1/2 in the closed disk to the circle of radius 1/3 in the open disk, then the circle of radius 1/3 in the closed disk to the circle of radius 1/4 in the open disk, and so on. Any point x in the closed disk that is not on one of these circles can just be matched up with the same point x in the open disk.