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MAT425 Real Analysis L4

Prof. Thistleton

Orderings

Cross Products, $A\times B$ .
Relations
We can define a relation from a set A to a set B as a subset of $A\times B$ .
We define a relation in A simply as a subset of $A\times A$ .
A relation on A, (denoted R ), is reflexive if $xRx\ \forall x\in A$ .
R is symmetric if, $\forall x,y \in A$ $xRy \Rightarrow yRx$ .
R is antisymmetric if, $\forall x,y \in A$ $xRy,\ yRx \Rightarrow x=y$ .
R is transitive if, $\forall x,y,z \in A$ $xRy,\ yRz \Rightarrow xRz$ .
An ordering on a set A is a relation which is reflexive, antisymmetric, and transitive.
Given a relation R defined on a set A we say two elements x and y are comparable if xRy or yRx.
An ordering on A is linear if any two elements of A are comparable.

Show that, with the definitions given above, the usual relation on the reals that we denote by < is an ordering of the real numbers but not a linear ordering.

Show that the relation defined on the reals and denoted $\le$ is a linear ordering.

The following definition is from our text. Is the preceding definition of a linear ordering the same as the one given in our text?

Linear Ordering
Let S be a set. We can define a linear order on S as a relation, denoted by <, such that

1.
If $x \in S$ and $y \in S$ then one and only one of the following is true:
$\begin{displaymath} x<y,\ \ \ \ \ x=y,\ \ or\ \ \ \ \ y<x \end{displaymath}$

2.
This relation is transitive. That is, if $x, y, z \in S$ and if x<y and y<z, then x<z.

Examples

Definitions

Let S be a subset of an ordered set. A least element of S is an element $x \in S$ , if one exists, such that if $y \in S$ and if y is comparable to x then $x\le y$ .
A linearly ordered set S is said to be well ordered if every nonempty subset of S has a least element.

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William Thistleton
11/10/1998