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equiv-stmts-in-fpg100

# equiv-stmts-in-fpg100 - Equivalence of Statements in the...

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Unformatted text preview: Equivalence of Statements in the FPG100 Geometry FPG100 Theorem 1: If the # of points is less than 100, then the # of points is an odd number. The FPG100 Geometry has 4 axioms: AX 1: There are finitely many points in the system. AX 2: There is only one line. AX 3: All points of the system are on the same line. AX 4: The number of points in the system is one of the following: 95, 97, 106, 108 . Proof: Suppose the # of points is less than 100. By AX 4, the number of points is 95, 97, 106, or 108. Since 106 and 108 are not less than 100, the number of points is 95 or 97. These are both odd numbers. Adding a fifth axiom may allow us to prove theorems _ _ The number of pomts IS an odd number. QED that we could not prove with the first four alone. Added Ax 5: Added Ax 5: The # of pts is The # of pts is less than 100. is an odd #. Thm: The # of pts is _ Thm: The # of pts is Is an odd #. less than 1 00. FPG100 AX 1; AX 2; AX 3; AX4 Added Ax 5: _ Added Ax 5: The # of pts Is The # of pts is not less than 100. is an even #_ Thm: The # of pts is not less than 100. Thm: The # of pts is is an even #. FPG100 Theorem 2: Proof of FPG100 Theorem 2 (continued) The statement "The # of points is less than 100" is equivalent to the statement "The number of points is an odd number." [Part II: Prove: it the # of points is an odd number, then the # of points is less than 100.] Suppose the # of points is an odd number. By AX 4, the number of points is 95, 97, 106, or 108. Since 106 and 108 are even numbers and 95 and 97 are odd numbers, the number of points is either 95 or 97. These are both less than 100. The number of points is less than 100. Proof: [Part I: Prove: if the # of points is less than 100, then the # of points is an odd number.] Suppose the # of points is less than 100. By AX 4, the number of points is 95, 97, 106, or 108. Since 106 and 108 are not less than 100, the number of points is 95 or 97. These are both odd numbers. QED The number of points is an odd number. [The proof continues to the right] Conclusion: In the FPG100 Geometry, the statement "The # of points is less than 100" is equivalent to the statement "The number of points is an odd number." ...
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