Math 3334: Class Notes

Definition 1 - The statement that P is a point means P is an ordered number pair.

Defintion 2 - The statement that Q is the plane means Q is the set of all points.

Definition 3 - Suppose each of x and y is a point. The statement that D is the distance between x and y means D= [(x₁- y₁)² + (x₂-y₂)²]^1/2 where X=(x₁, x₂) and Y=(y₁, y₂).

Definition 4 - The statement that R is a region means there is a positive number c and a point P and R is the set to which x belongs, only in the case x is a point and D(x,p) < c. We may denote such a region by R_c(P).

Problem 1 - if each of R and S isa region adn there is a point P common to R and S, then tehre is a region T such taht P is in T and every point of T is in each of R and S.

Definition 5 - Suppose M is a point collection. The statement that P is a limit point of M means P is a point and if R is a region containing P, then tehre is apoint of M, distance from P, that belongs to R.

Problem 2 - Every point of a region R is a limit point of the region R.

Problem 3 - If R is a region and P is a point not in R, then P is not a limit point of R.

Problem 4 - If M is a number collection that contains the number 1 and if n belongs to M, then n+1 belongs to M, does M contain all positive integers?

Definition 6 - The statement that M is an infitie point collection means if N is a positive integer then M has N points.