AN INTERPRETATION OF ROBERT ROSEN'S FUNDAMENTALS OF MEASUREMENT AND REPRESENTATION OF NATURAL SYSTEMS, North-Holland, NY, 1978

Seth Roberts, Author

Don Mikulecky, Editor and Media

CHAPTER 1: MATHEMATICAL BACKGROUND

[Note: For a brief introduction to set theory and related topics go to the Mathematical Physiology offerings under Instructional Materials on the Web]

The following examples interpret the material in chapter 1. They assume a basic familiarity with set notation and operations. See the above note if this is a problem.

Definition 1.1.1 The Cartesian Product

Let S = {1,2,3,4,5,6}, A set with six members which are integers.

The Cartesian Product of S with itself is then

S X S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

A Relation on S is a subset of S X S. Example:

Let R = {(1,1),(1,3),(1,5),

(2,2),(2,4),(2,6),

(3,1),(3,3),(3,5),

(4,2),(4,4),(4,6),

(5,1),(5,3),(5,5)

(6,2),(6,4),(6,6)}

Since R is a subset of S X S it is a Relation on S.

(4,2) Î R, \ we say 4R2 meaning that 4 is related to 2 via R.