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Partition the real plane into three subsets:

R, rational points - the set of all points, (X_r,Y_r), both coordinates rational (this constitutes the Rational Lattice);

I, irrational points - the set of all points, (X_i,Y_i), both coordinates irrational;

Q, quasi-rational points - the set of all points, (X_q,Y_q), one and only one coordinate rational.

Show also that any two points in the complete plane, (I unionQ unionR), may be connected by a path crossing no other rational points.

a. (Lemma) for any line on the plane with irrational slope, at most one rational point can lie on it. (Two rational points on the same line would define its slope to be rational.)

b. (Definitions) Any irrational coordinate identifies a path through the rational lattice, i.e. no line y=r_i or x=r_i, r_i an irrational number, goes through a rational point.

c. (Definition & Lemma) the set of all lines such that y=r_i or x=r_i, r_i an irrational number, constitutes the Non-rational Grid. The horizontal lines, all y such that y=r_i, intersect with all vertical lines, all x such that x= r_j.

Solutions and Sketch of Proof

1. All Irrational and Quasi-Irrational Points lie on the Non-rational Grid. Thus a path exists between any two such points which avoids all Rational Points.

2. For any rational point, A, a line with an irrational slope can be constructed which passes through the rational point. This line intersects the Non-rational Grid. Thus any other rational point, B, can be connected to A, without involving points other than A and B on the Rational Lattice.

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Comments:

a) Note that the Non-rational Grid is the complement to the Rational Lattice. The union of the two is the Euclidean plane.

b) the distinction between Irrational and Quasi-rational points could be done away with by calling the union of both sets, say, the "Non-rational" points. I maintained the distinction, however, because the original problem was formulated in terms of it.

c) What does this mean for those graphs of "indifference curves" one finds in economics? Assuming that commodity "bundles" can always be approximated by a rational number, then there is no reason to work with the real plane. But the rational lattice does not automatically yield indifference curves even though some commodity bundles may be indiscriminable by either party to the bargaining situation. The solution to this brain-teaser indicates that there is always some connected path that lies outside the rational lattice. It would seem then that connectivities on the rational lattice have to be constructed by something other than assumptions about the numbers that commodity points are expressed in.

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