We will use to denote the set of real numbers and to denote the set of natural numbers. Sometimes real numbers are called scalars. Whenever A is a set, we will write to say ``x is an element of A.'' If A and B are sets we will write to say ``A is a subset of B.'' If A and B are sets then we define
We say that a set A is finite if there is a natural number n and elements in A for every natural number so that
For example, the set is a finite set, but is not a finite set.
and
Show that if and then
and
and
(Note: The argument above is given in radians, not degrees!!)
Prove your claim.
Let P and J be points in . We define
Then, inductively, for n=2,3,... define
Similarly, we define
B1={J}
Then, inductively, for n=2,3,... define
Let and Show that Notes: is the midpoint of the line segment joining U and V. The choice leads to the set A shown below.
Now create a set A as follows. Let be a point in Set
and inductively, for n=2,3,4,... define
Then set