Logic in Elementary Mathematics by Robert M. Exner & Myron F. Rosskopf

Author:Robert M. Exner & Myron F. Rosskopf
Language: eng
Format: epub, mobi
Publisher: Dover Publications, Inc.
Published: 2013-04-16T16:00:00+00:00

FIG. 21.

Proof: Call the point “P”. P joins just three lines [A2*], call them “l1”, “l2”, and “l3”.

l1 has exactly two other points on it [A2]. Call them “X1” and “Y1”. X1 is joined to l2 at P [hyp]; so X1 is joined to exactly one other point on l2 [A5], call it “X2” (see Fig. 21).

Similarly, X1 is joined to some point of l3 other than P. Call it “X3”.

Similarly, X2 is joined to a point on l3 different from P, call it “F” (Fig. 21).

We would like to prove that F = X3 so that (X1X2X3) is a triangle. We proceed indirectly.

Suppose X3 ≠ F.

Now, the figure suggests that there are three lines joined by X2. Let us prove that they are indeed all distinct. If (X1X2) is the same line as l2, that is, (X1X2) = l2, then X1 is on l2 and l1 = l2 [T1]. But we have l1 ≠ l2, so we have (X1X2) ≠ l2 [contrap. inf.].

Similarly, (X2F) ≠ l2.

Finally, if (X1X2) = (X2F), then P is joined to all three points of this line, which contradicts “A5”. Thus, there are three distinct lines (X1X2), (X2F), and l2 joined by X2. By “A2*” these are the only lines on X2.

Now,

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