Axiom 1. S is a collection of points such that: A. if x and y are different poin

Question

Axiom 1. S is a collection of points such that:
A. if x and y are different points, then x<y or y<x.
B. if the point x precedes the point y, then x is not equal to y.
C. if x, y, and z are points such that x<y and y<z, then x<z.

Axiom 2. S has no first point and no last point.

Problem 3.1. Suppose “point” means a point on the unit circle. Show that there is a meaning of “precedes” such that both axioms are true.

Problem 3.2. State in words, and draw, what a typical region would look like in each of the following three models:

• Where S is the set of integers;
• Where S is the set of real numbers;
• Where S is the set from Problem 2.3 (PROBLEM 2.3: (Lexicographic Interpretation) Suppose by point we mean ordered number pair. What interpretation could you give to the word precedes so that Axiom 1 is satisfied? )

Explanation / Answer

13.1

The equation of unit circle is given as,

x2 + y2 = 1

Let (0, 1) be the point on the unit circle.

Now, 0 precedes 1 and hence 0 is not equal to 1.

So, the preceding condition of axiom is satisfied.

Also, the collection of points on the unit circle has no first and last point.

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