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Lesson 1: Axiomatic systems and Incidence Geometry Outline Axiomatic systems Models of axiom systems An example of Axiom system Parallel postulates Conditional propositions and proof Example Let D and E be the midpoints of the side AB and AC of ABC respectively. Then DE|| BC. Questions: 1. What does this sentence mean? 2. What is the midpoint? 3. What does ABC mean? 4. What is DE|| B ? 5. Is the statement correct? Why ? 1.1 . Axiomatic system 1.1.1. Example. In Euclidean geometries, we have the following: Point, lines, perpendicular lines, triangle, angle, rectangle, similar triangle, “ It is possible to draw a straight line through any two given points” “It is possible to construct a circle with any given center and radius” “ Any two right angles are equal” 1.1.2. Main components of an axiomatic system Undefined and Defined terms Undefined terms are the technical words that will be used in the subject. Defined terms are the words that are defined or described using undefined terms and those have been defined previously. Axioms ( postulations) The axioms are statements that are accepted without proof. The axioms are given meaning to undefined terms. Theorems ( propositions) Theorems are the statements that can be proved from the axioms or proven theorems. Models An interpretation of an axiom system is a particular way of giving meaning to the undefined terms in that system. A model of an axiom system is an interpretation of the system such that all the axioms are corrected statements in that interpretation. 1.1.3. Example Euclidean geometry, Undefined terms: point and straight line are undefined terms. Postulations: 1). Any two points can be joined by a straight line. 2). Any straight line segment can be extended indefinitely in a straight line. 3). Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4). All right angles are congruent. 5). If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The Postulation 5) is equivalent to the parallel postulation in this geometry: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. l A 1.1.4. Independent statements, consistent system A statement is independent of the axioms if it not possible to prove or disprove it from the axioms. A good way to prove the independence of a statement is to find one model in which the statement is true and another model in which the statement is false. An axiom system is called consistent if there are no two contradict theorems in the system. If there is a model for the system, then it is consistent. 1.2 . An example of an axiomatic system( Incidence geometry ) Undefined terms: point, line, lie on (incident) Definition Three points A, B and C are collinear if there exists one line l such that three of the points A, B and C all lie on l. The points are noncollinear if there is no such line l. Axioms (IA 1) For every pair of points P and Q there exists exactly one line l such that both P and Q lie on l. (IA 2) For every line l there exist at least two distinct points P and Q such that both P and Q lie on l. (IA 3) There exist three noncollinear points. 1.2.1. Example The three-point plane model. Interpretation: a point means one of the symbols A, B and C a line is one of the sets {A, B}, {A,C} or {B,C}. “ lie on “ means “ is an element of “ Exercise: Check that all the axioms of incidence geometry are satisfied. B A C 1.2.2.Example Points: A, B,C Line: {A, B, C} Which axioms are satisfied? 1.2.3. Example Fano’s geometry C Points: A, B, C, D, E, F, G B Lines: A G {A,B,C}, {C,D,E}, {E,F,A}, D {A,G,D}, {C,G,F}, {E,G,B}, F {B,D,F} E 1.2.4. Example The Cartesian plane A point is a pair (x,y), x,y are real numbers A line is a set of points whose coordinates satisfy a linear equation of the form ax+by+c=0, where a,b and c are fixed real numbers for the line and a and b are not both 0. A point is said to lie on a line if the coordinates of the point satisfy the equation. 1.3. Parallel postulates 1.3.1.Definition Two lines l and m are said to be parallel if there is no point P such that P lies on both l and m. If l and m are parallel, we write l || m. 1.3.2. Three Parallel postulates Euclidean Parallel Postulate: For every line l and for every point P that does not lie on l, there is exactly one line m such that P lies on m and l || m. Elliptic Parallel Postulate: For every line l and for every point P that does not lie on l, there is no line m such that P lies on m and l || m. Hyperbolic Parallel Postulate: For every line l and for every point P that does not lie on l, there are at least two lines m and n such that P lies on both m and n and m and n are both parallel to l . 1.3.3. Example (a) The three-point plane does not satisfy Euclidean’s Parallel Postulate. It satisfies the Elliptic Parallel postulate. (b) The four-point geometry satisfies the Euclidean Parallel postulate. Points: A,B,C,D Lines: {A,B},{A,C},{A,D},{B,C},{B,D},{C,D} C D A B (c) The Cartesian plane satisfies the Euclidean Parallel Postulate. Exercise: Consider the five-point model. Points: A,B,C,D,E Lines: All pairs of points Lie on means “ is a member of “ Which Parallel Postulates does the five-point model satisfy? The parallel postulates is independent of the axioms on incidence geometry. Summary: 1. An axiom system is determined by some undefined terms, axioms . 2. A theorem is a statement that can be proved from the axioms or other theorems 3. A model of a system is an interpretation of the system so that all the axioms of the system are corrected statements in that interpretation. 4. Incidence geometry 5. There are three different Parallel postulations: Euclidean Parallel Postulate: Elliptic Parallel Postulate Hyperbolic Parallel Postulate: Now try Exercise set 1. 1,2,3,4 1.4. Conditional propositions 1.4.1. Propositions A proposition in mathematics means a statement which is either true or false but not both. (1) The addition of two even numbers is an even number. (2) x+2<5. (3) If y<5, then y+2<7. (4) Goldbach conjecture: If m>2 is an even integer, then there are prime numbers p and q such that m=p+q. (5) If x< 5 and y<7, then x+y<20. 1.4.2. Conditional propositions: If l is a line, then there exists at least one point P such that P does not lie on l. Hypothesis Conclusion H C C H true true true true false false false true false false false true 1.4. 3. Example A father told his son: If you get an A in your next math exam, I will buy a new lap-top for you. In which case(s) do you think the father breaks his promise? 1.5. Inverse, converse and contrapositive For any given proposition p, the negation p of p is the proposition which is true only when p is false. From the proposition p q , we can form three new propositions: Converse: qp Inverse: p q Contrapositive: q p Every conditional proposition is equivalent to its contrapositive. 1.5.1. Example Let m and l be two distinct lines. If m and l are parallel, then there exists no point that lies on both m and l. Converse: If there exists no point that lie on both m and l, then m and l are parallel. Inverse: If m and l are not parallel, then there is a point that lie on both m and l. Contrapositive: If there exists a point that lie on both m and l, then m and l are not parallel . Exercise: State the inverse, converse and contrapositive of the following proposition: If ABCD is a parallelogram, then the two diagonals AC and BD bisect each other. 1.6. Proof of a proposition 1.6.1.Example If l is a line, then there exists a point P such that P does not lie on l. Proof 1. l is a line ( Assumption) 2. There exist three distinct points P , Q and R that are not collinear ( By Axiom 3) 3. At least one of P, Q and R does not lie on l ( By 2) 4. There is a point that does not lie on l. A proof of a proposition consists of a sequence of statements each of them is either an assumption , or an axiom, or a statement derived from the previous statements. The last statement is the conclusion. Proof 1. Statement ( reason) 2. Statement (reason) 3. … … m. Conclusion (reason) 1.6.2. Proof by contrapositive Since the proposition “ If P then Q” is equivalent to its contrapositive , we can prove “ If P then Q” by proving its contrapositive “ If not Q then not P”. 1.6.3. Example If m and n are integers and mn is an even number, then at least one of m and n is even. Proof 1. Suppose both m and n are odd numbers ( Contrpositive assumption) 2. m=2p+1, n=2q+1 for some integers p and q ( by 1 ) 3. mn=(2p+1)(2q+1) =2(2pq+q+p)+1 ( By 2 ) 4. mn is odd ( By 3 ) Exercise Prove the following by contrapositive: If x and y are two integers whose product is odd, then both x and y must be odd. 1.7. Indirect proof ( Proof by contradiction) To prove “ If P then Q” by contradiction we assume P is true and Q is not true and derive two contradict propositions. To prove “ If P then Q” by contradiction we assume P is true and Q is not true and derive two contradict propositions. Example If m and l are distinct , nonparallel lines, then there exists a unique point that lies on both m and l. Proof 1. m and l are distinct lines, m and l are not parallel. (Assumption) 2. there exists a point P lie both on m and l ( by 1 ) 3. suppose there is another point Q that also lie on both m and l. 4. m and l are the same. ( By 2, 3 and Axiom1) 5. so P is the only point that lies on both m and l. Thank You! Submit the solutions of Exercise Set 1 Q4, Q5 , Q 8 by 22 Jan 2008

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