9
Number Systems
In this chapter, we will review the basic concepts in real analysis: order re-
lations, ordered sets and fields, construction and properties of the real and
the complex fields, and finally the theory of countable and uncountable sets
together with the cardinal numbers. The known sets of numbers that we will
use in this chapter are- N: Natural
- Z: Integer
- Q: Rational
- R: Real
- C: Complex
9.1 Ordered Sets
Definition 9.1.1 Let S be a set. An order on S is a relation -< such that
i) If x, y are any two elements of S, then one and only one of the following
is true:
x < y, x = y, y < x.
ii) If x,y,z G S and x <y and y -< z, then x -< z.
x <y i^ y <x.
x < y means x -< y or x = y without specifying one.Example 9.1.2 S = Q has an order; define x -< y ify — x is positive.
Definition 9.1.3 An ordered set is a set S on which there is an order.
Definition 9.1.4 Let S be an ordered set and 0 ^ E C S. E is
- bounded above if3bES^VxEE,x<b where b is an upper bound of E.
- bounded below if 3a G S 9 Vx £ E, a -< x where a is a lower bound of E.